I've read multiple physicists talk about stuff (e.g. galaxies, black holes, etc.) "existing" outside the observable universe, but it's unclear to me in what sense these things "exist". By definition things outside the observable universe are spacelike separated, hence casually disconnected, from us. And since two spacelike separated events can always be made simultaneous by taking the right reference frame, we can always pick a reference frame in which any given event outside the observable universe "hasn't happened yet". It seems to me the more accurate thing to say is that the observable universe is "growing" and that "outside the observable universe" is a collection of events that "haven't happened yet". What do physicists mean when they talk about things existing outside the observable universe?
I've read multiple physicists talk about stuff (e.g. galaxies, black holes, etc.) "existing" outside the observable universe, but it's unclear to me in what sense these things "exist".
You ask what the meaning of "exist" is when it is not observable, or more specifically in this case, spacelike separated. I assume that you are not going down the purely empiricist worldview when you ask this question, and are instead talking about the definition of "exist".
I see no problem with the existence of objects that are spacelike separated from me. I honestly question whether anyone does. Sure, one cannot make observations about specific objects, you know, that black hole or that galaxy, but does anyone really think that galaxies can't exist beyond the cosmological horizon, or that even asking that is somehow a non-sequitur?
From a physical standpoint, inflationary cosmology suggests that the cosmological horizon is purely an observational construct, and has no physical basis. Adding that to big bang cosmology results in the statement that the physical universe is dramatically larger than the observable, but it all has a common physical basis. For that reason, there is a very good reason to believe there are objects just like our own on the other side of the imaginary sphere.
There may be some conflation of "event" and "exist", which may be reasonable in GR but I'm not sure works in a wider context.
IMO there is confusion (also in the comments) between objects, things and events. To me is meaningless to say that an event exists. It may be spacelike separated from myself writing this comment, but become - the event - timely separated tomorrow, then observable.
When we speak of "observable universe" we aren't thinking of events but of objects (e.g. galaxies) which have a continuous existence in time and in some past time could have emitted light now arriving to us.
You might usefully consult wikipedia for "cosmological horizon".