Delayed Choice Quantum Eraser without retrocausality? After reviewing the more than half dozen questions on the DCQE here, this doesn't seem to have been asked or answered directly - if I've missed this piece in one of the other questions or answers please point me in the right direction and we can close this.  
Using the detector notation from the DCQE Wikipedia article for brevity:
Let $D_0$ = idler photon collector plate
Let $D_1$ = no-path information detector 1
Let $D_2$ = no-path information detector 2
Let $D_1$ and $D_2$ be time-like separated from $D_0$ such that $D_0$ will register photon arrival hits some finite, measurable time prior to $D_1$ or $D_2$.
For the purposes of this question, we can ignore the cases where which-path info is collected.  Frankly, we can ignore the "choice" to insert a beam splitter or not entirely.  For the purposes of this question, a half-silvered mirror will remain in place (no which-path information is available).
After many runs, when I go back to correlate the arrival times of clicks at $D_0$ with clicks at $D_1$ and $D_2$, I note that 2 interference patterns are recoverable, with the location of $D_1$ idler entangled photons aligning with troughs of $D_2$, just as expected.
Since I now know the locations of the respective peaks and troughs on $D_0$ correlated with $D_1$ and $D_2$ detection, I wipe my detection plates clean and start again.  I note that my first "click" at $D_0$ is in a location that corresponds to a peak in the $D_1$ fringes.  
My question is simple:
Based on this knowledge, is the signal photon more likely to be detected at $D_1$? (my $D_1$ detector hasn't clicked yet due to the time-like separation).
It seems the answer must be yes.  
If it IS yes, then why is there any discussion at all about backwards causation?  If I can note the location of photon recording on $D_0$ and predict a higher or lower probability of recording an entangled sister photon on $D_1$ or $D_2$, then that's the ballgame. Where's the spookiness that has this experiment come up again and again?
If the answer is no in this setup, I struggle to see why, and so any clarification would be greatly appreciated.  
 A: The short answer is that you're right. The delayed choice experiment doesn't require backwards-in-time shenanigans, and all the pop-science articles implying this to be the case are basically garbage.
For example, here is the example delayed choice eraser circuit from my quantum circuit simulator Quirk:

The green boxes are state displays that show the probability of each possible measurement result, optionally conditioned on some other qubits' possible measurement results.
The top wire is the "choice" qubit. The second wire is the "which slit" qubit. The rest of the wires are the "where did it hit on the screen" qureg. The first two operations setup some entanglement between the which-slit qubit and the screen qureg.
The four displays on the right show that, if you group the screen measurements by the choice qubit and the which-slit qubit, then within the groupings you will see an interference pattern if and only if the choice qubit is ON.
But correlation goes both ways. Instead of thinking about how the choice qubit and which-slit qubits predict the screen measurement, we can think about how the screen measurement predicts the which-slit qubit's state.
It would be tedious to set up $2^7$ bloch sphere displays, each conditioned on a different screen measurement. Instead, let's use a single condition but cycle through offsets to the screen measurement. This circuit makes it very clear that the landing position on the screen is correlated with different states of the which-slit qubit:

This diagram also answers your main question:

[given a landing position] is the signal photon more likely to be detected at D1 [than D2]?

Yes. The value analogous to the D1-vs-D2-given-position likelihood is shown in that green box in the top right of the diagram. The chance is changing as we focus on different positions.
Notice that the qubit is spinning like crazy regardless of whether we applied the $X^{1/2}$ rotation that will be controlled by the choice qubit. All the $X^{1/2}$ rotation does is switch which axis the qubit is spinning around as I vary the screen measurement position being focused on. If there is no $X^{1/2}$ rotation, the spinning goes around the measurement axis, and so doesn't affect the probability of measuring ON-vs-OFF (i.e. we've picked a rather poor measurement axis). But if the $X^{1/2}$ is applied, then the spinning is around the Y axis instead of the Z axis, and so does translate into changes in computational-basis measurement probability.
So here is a forward-in-time collapse interpretation of the delayed choice experiment:


*

*We create a situation where the which-slit qubit is entangled with a screen-landing position.

*We measure the screen-landing position. This collapses the system into a state where the which-slit qubit is pure, but the specific direction it points is determined by the landing-position measurement.

*We choose to measure along an axis that picks up these variations-from-landing-measurement, or to measure along a totally uninformative axis.

*We act surprised that our choice determined whether we see correlations between the which-slit qubit measurement and the screen landing-position measurements.

*We switch to thinking about the correlations in the opposite direction and start talking about time travel, because that's what brings in the sweet sweet clickbait ad revenue.

A: According to your description I agree with you that a prediction, that idler photon is more likely detected at D1 or D2, based on knowledge where signal photon hit D0, is possible.
In your simplified experiment a 50% beam splitter (BSc) has remained.
I think that reason why discussion about retrocausality has started, is the following:


*

*A 50% beam splitter is a device which reflects 50% of photons, the other 50% can pass through. So in case of a laser beam which consists of a huge amount of photons, it is predictable that 50% of the photons will be reflected.

*BUT it is impossible to predict for an individual photon, whether it will be reflected or not. This is pure objective coincidence.


Now, according to your description it is possible to do a prediction for a single photon (the idler photon) which is arriving at BSc (some nanoseconds after entangled signal photon hit D0), whether it will be reflected at the splitter or not.
This would be an astonishing result as such a prediction should not be possible at all.
Retrocausality comes into picture if such a prediction of an individual event is unacceptable and hence still considered as impossible. In such case, idler photon would unpredictably hit at D1 or D2.
As a logical consequence, result at D0 would have to be changed retrocausal, so that proper interference pattern would appear at the end.
A: This answer makes the assumption (unwarranted as of v5 of the question), that you're using the notation from the Wikipedia article on the topic.
Given that, in the situation you have described, the answer to your core question,

is the signal photon more likely to be detected at $D_1$?

is yes. If the interference were perfect (i.e. in a noiseless experiment), if you detect a photon in a position where the $D_0\wedge D_2$ coincidence rate has a node, then you can be absolutely certain that the idler will be observed at $D_1$ and not $D_2$. If the experiment is less than perfect, or the detection is slightly away from one of the nodes, then there will be a small probability of detecting the idler at $D_2$ but this will still be smaller than at $D_1$.

Now, as you note, if the $D_1/D_2$ detection is in the causal future of the detection at $D_0$, and in the absence of the possibility of introducing which-way detections, then as far as causality goes the experiment is less challenging, and there are reasonable classical models (which are specific to this experiment, and not extensible to a general theory) that do a good job of describing the dynamics without any weird quantum fuzziness.
However, that's just because you've artificially de-fanged the experiment by removing the core part of the argument ─ the choice in delayed-choice quantum eraser. The standard argument posits that within the data collected at $D_0$ there are already latent interference patterns even though the photons were 'marked' with information that can be used to discern which-path information after the (potentially) interfering photon has already been detected. That seems to bother some people.
In particular, let's take your analysis and run with it: suppose that you've measured the photon at one of the nodes of the $D_0\wedge D_2$ interference pattern, and therefore you're certain that if you measure on the $D_1/D_2$ no-which-way pair then you'll go always see it on $D_1$. Then some enterprising folks might say "aha! now that we know it's interfering, and how, let's not do the $D_1/D_2$ pair, let's do the $D_3/D_4$ which-way pair instead" ─ and, of course, they find out that once they post-select on that the interference patterns disappear.
I agree with you that the delayed-choice quantum eraser is not particularly surprising (or at least, I don't see how it adds anything much beyond the standard quantum-eraser configuration), but you're cutting out key bits of the analysis, and that rarely bodes well.
A: With the caveat that I don't really understand the views I'm about to describe, I'm going to try and explain where talk of retrocausality comes from.
Historically, there had been a lot of resistance to the picture of reality painted by quantum mechanics, and people were looking for less well-defined but more palatable ways to explain experimental behavior. (with, of course, the hopes of eventually discovering a well-defined theory)
The more famous experiments are about killing these various alternative ideas. For example, the various Bell tests that killed local hidden variable theories.
The main idea at play here, I think, originates from the basic double slit experiment. You normally see an interference pattern, but if you add something to the experiment to discern which-way information, the interference pattern goes away.
The potential idea that spawned is that the photon is normally in some "I will interfere" state, but the presence of which-way apparatus tells the photon to switch to an "I won't interfere" state.
This may sound squirrelly, but remember that this isn't in the "I have a concrete idea that makes precise, testable predictions!" phase of science, it is in the early stages of the "I am trying to figure out a way to make sense of the data!" phase of science.
The quantum eraser experiments were a big step towards killing these ideas; despite the eraser afterwards, the which way apparatus is still there to spoil the interference. But these ideas can still be rescued in various ways; e.g. by positing that it still takes some time for the which-way apparatus to spoil the interference, or the presence of the eraser tells the photon not to switch to the "I won't interfere" state.
These attempts at salvage suffer a mortal blow from the delayed choice quantum eraser experiments; the photon is forced to "choose" before the choice to erase has been made, so any attempts to stick to these sorts of alternative explanations are forced to resort to retrocausality.
A: Old question, and the physical part of it has already been explained sufficiently by the other answers. I'd like to answer your question:

If it IS yes, then why is there any discussion at all about backwards
causation?

The main reason is because people do not understand that the pattern at D0 does not actually change. People who have heard of the experiment, but have not actually looked at the details of it, think that which D1-D4 detector the photon lands determines where its twin lands (has landed) at D0. The D1/2 photons show an interference pattern at D0, while the D3/D4 photons show a bulge pattern at D0, so where the D1-D4 photon lands must influence where its twin lands at D0, right? I think it is an easy misinterpretation to make by a layman, and it even happens to respected presenters like PBS Digital Studios: https://www.youtube.com/watch?v=8ORLN_KwAgs&t=157s.
They don't understand that all photons always show interference at D0, and nothing changes in that based on what happens at the other side. It is just that if you add up the interference patterns of the photons that land at D3 or D4, that they happen to add up to form a bulge pattern.
