My reasoning being that due to Lorentz invariance, if two objects are located at time-like separated spacetime points, then it is possible to find a frame in which they are located at the same spatial point, in which case any direct interaction between the two objects would violate causality (one object would [instantaneously] affect the other despite the temporal separation between them).
This is an inappropriate use of the word instantaneous. It is possible for the earlier one to indirectly affect the later one, but the later one does not affect the earlier one at all. Instantaneous means happening at the same time (the same instant, which is a hyperplane of simultaneity).
If you don't want the future to affect the past then it is not a symmetric interaction of each influencing the other.
Next, if two objects are located at space-like separated spacetime points, then it is possible to find a frame in which they are located at the same point in time, in which case any direct interaction between the two objects would constitute action-at-a-distance which is undesirable as it suggests that spatially separated objects can directly affect one another (i.e. without any discernible mediation of the interaction between them), regardless of the distance between them.
This would be instantaneous. And your objection sounds tautological, even though you are bringing up different frames. This objection to space-like separation sounds like an objection to non locality, I don't see any explanation.
There are legitimate concerns. If different people disagree about which happens first it is hard for them to affect each other without the future affecting the past, so it is about the relativity of simultaneity. Many people cite that as the objection. But calling events simultaneous is just a label, as if future or past (as opposed to proper future and proper past) is an important distinction even though it is an entirely unphysical frame based distinction (whether two spacelike separated events have one in future or past or neither of the other).
These events are in the elsewhen (neither the proper future nor the proper past) of each other.
The argument of mismatched simultaneity at least builds on the previous objection (of not wanting the future to affect the past) but there is a more objective and more practical experimental concern. Specifically about predictability, reproducibility and control.
We want to make predictions. We want to test the predictions. We want the tests to come out correct reliably. That means a prediction has to predict a future situation that can be measured but we need to explicitly state how that future situation depends on current things that can be controlled or measured.
So the real problem is one of access. We don't have access to the future when making a prediction. And if we want to make predictions about continuously unfolding of dynamical systems then our predictions can be for very small time intervals later than now. Once something is located somewhere else (spacelike separated, so located elsewhen), then to some observers those events are happening a fixed time later so for intervals smaller than that it is part of the future you don't have access to.
Not having access to the future when making predictions now is the reason that timelike and spacelike events don't have one affect the other if there is a frame where one happens later.
Finally, if two objects are located at light-like separated spacetime points, then it is impossible to find a frame in which they are located at the same spatial point or temporal point, in which case any direct interaction between the two objects implies action-at-a-distance and a violation of causality.
Same thing here. If they are lightlike separated then you can choose a time interval small enough where one event comes later than the other so is not available to make predictions about how things are changing now, which is what we do.
We want locality because we don't want to say that our results are determined by someone far away and how they adjust some knobs in their equipment. But even that isn't the reason for local interactions.
The real reason is symmetry and conservation laws.
Continuous symmetries lead to conservation laws. And we want symmetry because otherwise things would just be different at different times or places for no reason whatsoever. They can be different because they can depend on the things around them, but they shouldn't depend on some global clock tick or some global x axis or some global origin. And this is a kind of locality, but it is the true locality we want that everything else is built from.
So, once we have the symmetries, we get the conservation laws. And the conservation laws allow to you asked yourself how much energy and momentum there is. And here is the conclusion, the energy and momentum have to flow from two things at the exact same location because if something loses energy at one event and that energy is gained at a different event then there is a hyperplane of simultaneity that divides the two events and two hyperplanes parallel to that one that go through each event and the energy and momentum are not the same on the three hyperplanes because that middle intermediary one one is missing that energy.
So there is one kind of locality, a symmetry, and it leads to the fact that interactions happen at the exact same event.
And there is no breakdown of spacelike, timelike, or lightlike. Just same event versus different event. And this is the actual perspective of actual physicists.
[A] cause at a particular spacetime point should not have a direct effect at other spacetime points, regardless how small the separation
Causality is a tricky beast. In general, you can have say an electromagnetic field and its temporal rate of change matches some combination of its spatial deviations. For a charged particle it's the second temporal derivative that depends on what is going on (for instance on the spatial variation of the potentials).
But we can assign energy to the fields and energy to the particles and the point is that the events when-where the field gains or loses energy are the same events when-where the particle loses or gains energy. And anything else would violate conservation of energy.
There doesn't seem to be a lot of literature on this topic though.
I have seen multiple whole books about nothing else. And various articles. And different theories have different takes. For instance the Feynman-Wheeler electrodynamics had action at a distance interactions between lightlike separated events but it doesn't have conservation of energy.
Since symmetries require energy conservation and energy conservation requires local interactions at the exact same point that is the real reason people want it. But some people just want it to be local so stop there. Some want energy conservation and so they need local interactions. Other people want the symmetry and so are forced to have the conservation so are forced to have the locality.
But I've argued that the symmetry isn't really different than the locality, it's just a preferred form of locality. The symmetry required that you not care about a global origin, only about the things around you, so that's the preferred fundamental locality for most people.
