Therefore, grand pronouncements about how the universe is "really" a static four-dimensional object "because that's what relativity says" are just wrong, for the exact same reasons.
Perhaps even in the "not even wrong" sense, on the grounds that it is no different than taking Newtonian physics and making ground pronouncements about the nature of the universe. Newtonian physics implies things like "there can be no absolute speed limit in the universe"; not just that there isn't one, but that there can't be one for the transforms it uses to be valid. There is such a speed limit and the transforms it uses aren't valid. Declaring the universe to have this or that characteristic based on relativity is no different than declaring it must not have a speed-of-light because that's what Newtonian physics says. The only difference is that "everybody" knows the latter is wrong; the former is just as wrong.
So in general getting too worked up over what Einstein's relativity says about the universe at this level is a waste of time, no pun intended.
Science history being what it is, it is quite likely that if we ever do penetrate down to what time "really" is it'll be even more mind-blowing than a static four-dimensional universe, but that's a problem for the future. (And the current leading contenders in that theory race I'm not sure are any more disruptive than QM already was. Total chaos space/time at scales so small that they are in some sense literally microscopically microscopically microscopically microscopic may be vaguely unsettling, but to my mind doesn't seem to add anything philosophically material that QM didn't already introduce.)