Einstein's goal was to demonstrate that quantum mechanics was inconsistent by coming up with a thought experiment where it would be possible to measure an electron's position and momentum simultaneously. Bohr and Heisenberg's goal was to show that quantum mechanics was ok, because Einstein's thought experiments inevitably assumed a violation of the uncertainty principle, usually in some subtle place.
From a more modern point of view, where we accept quantum mechanics gives a good description of reality, these thought experiments are mostly of historical interest, and can be confusing to learn from because they mix classical and quantum ways of thinking. However, it is also interesting to think about them.
The notion of "collapse" is part of our modern understanding of quantum mechanics (at least if you buy the Copenhagen interpretation). It is not something Einstein would have accepted. So while we can make sense of this thought experiment using the idea of collapse, Einstein would not have accepted an argument using the concept of collapse.
Below, I'll outline two different ways to think about this experiment, one using a more modern point of view where we can use the concept of "collapse", and one more historical point of view where the goal is to show that Einstein has violated the uncertainty principle in his thought experiment, meaning his thought experiment does not prove quantum mechanics is inconsistent (since assuming the uncertainty principle is violated amounts to assuming quantum mechanics is incorrect, so he assumed what he was trying to prove).
Using the collapse language -- you are completely correct. The interference pattern only appears if we do not collapse the particles before we measure them on the screen on the far right hand side of the diagram. If we measure which path the particles take, therefore causing the state of the particle to collapse to one path or the other, then we will see the particles landing in one of two locations, rather than in an interference pattern.
Using the uncertainty principle language -- as Feynman says, in order to measure the difference in momentum of the plate after the particle passes through, we need to know the initial momentum of the plate before the particle arrives. And, as Feynman says, Einstein snuck in his assumption that the uncertainty principle was violated by assuming he could both measure the plate's momentum to be exactly zero, and know the position of the slits.
Strictly speaking, that's enough to show what we wanted to show -- we've already shown that Einstein's construction cannot show quantum mechanics is inconsistent, because he started off making an assumption inconsistent with quantum mechanics.
However, we can actually go further, and show how fixing this assumption gives a result consistent with quantum mechanics. This upgrades our argument from correct (Einstein hasn't shown quantum mechanics is inconsistent because he made a circular argument) to devastating (Einstein isn't going to be able to fix his argument in a simple way because the uncertainty principle introduces precisely the right amount of uncertainty to resolve the issue he raised). If we account for the uncertainty principle, and we assume that we know the initial momentum of the plate is exactly zero (or at least very close to zero), then we must have a large uncertainty in the position of the slits. As Feynman says, this uncertainty in the position of the slits means that the possible particle trajectories will spread out. This spreading out will kill the interference pattern. The interference pattern is a delicate thing that requires cancellations between well-defined paths -- adding in a coarse averaging over different possible paths the particles could take because we don't know where the slit was will destroy these cancellations. Then the particles will appear in one of two lumps, which is what we would expect of classical particles. This result is also completely consistent with the "collapse" language used earlier -- these are two different ways of talking about the same phenomenon. The collapse picture is probably easier to follow, so long as you accept the idea of collapse, whereas the less direct approach in this paragraph is perhaps trickier to follow, but doesn't use concepts Einstein would reject out of hand.
Mathematically establishing the statements I made in my last paragraph, is what Feynman claims he will do in a future chapter.
