I've recently watched PBS SpaceTime's video about the quantum eraser. In it, Matt describes a kind of crystal that splits particles into pairs and entangles them. The entangled particles are then delivered into the quantum eraser part of things. My question is, if we remove all the detectors and half-silvered mirrors and whatnot and let the other entangled particle fly off into space, undisturbed and undetected, would an interference pattern still emerge since we would not obtain the path information?
No, an interference pattern will not appear.
The key criterion whenever which-way information is involved is whether the information is available in principle, regardless of whether we detect it or not. The presence of the half-silvered mirrors and the detectors A and B (in PBS Space Time's notation) is irrelevant: the photon flying away is entangled with the photon that hit the screen, and in that entanglement it carries the which-way information of what slit both went through.
The only way to restore the interference pattern is to completely erase the which-way information, in a way that makes the reconstruction of which slit the photons went through impossible even in principle. The standard way of doing it is the one explained in the video (i.e. by making a coherent measurement between the two options, and then post-selecting out on the both outcomes, which will separate out the two overlapping complementary interference patterns on the screen), but in general the procedure needs to be coherent, it needs to perform a nontrivial projective measurement on the existing which-way information, and it needs to do a correlated post-selection to get anything meaningful.
If you don't do any of that (say, if you just let the which-way information fly away) then the interference pattern won't come back.