How do we know the assumptions of the Schwarzschild solution are valid? The Wikipedia article on the derivation of the Schwarzschild solution (https://en.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution) lists 4 assumptions.  The second of which is:

A static spacetime is one in which all metric components are independent of the time coordinate $t$ (so that $\tfrac\partial{\partial t}g_{\mu \nu}=0$) and the geometry of the spacetime is unchanged under a time-reversal $t \rightarrow -t$.

How do we know that this assumption is physical?  How do we know that spacetime doesn't invert under a time-reversal instead?  Is there any experimental data to support it?
 A: The spacetime be static is not a necessary assumption for obtaining the Schwarzschild solution. Birhoff's theorem tells us that any spherically symmetric vacuum solution to the Einstein equation is also static and asymptotically flat, and therefore must be (part of) the Schwarzschild solution.
Of course as Andrew's answer explains physical solutions cannot be vacuum everywhere, and probably are not exactly spherical either. The Penrose-Hawking singularity theorems, however, tell us formation of black holes will happen generically. The No hair theorems ensure that after collapse happens the result will asymptote to a member of the Kerr family of solutions.
Consequently, working with the Kerr solution (or its special non-spinning) case as actually a reasonable physical assumption when dealing with a black hole that has had sufficient time to settle down.
A: This is a very long-winded answer, so let me state the punchline up front: the boundary condition you pointed out really is physically suspect, and probably not realized in Nature -- however using it anyway leads to a very useful exact solution of Einstein's equations!

The full Schwarzschild solution (sometimes called the maximally extended Schwarzschild solution) is the metric which solves Einstein's equations subject to the boundary conditions you cite. The solution actually describes 4 interconnected regions of spacetime, as can be seen in this diagram (which I took from https://jila.colorado.edu/~ajsh/insidebh/penrose.html):

First, a note on the rules for how to read this diagram.

*

*Time flows from bottom to top.

*"Left/right" motion is essentially "radial" motion toward or away from the center of the spacetime.

*The diagram is 2 dimensional but spacetime is 4 dimensional; in fact, each point on this diagram represents a 2-sphere at the same time and radius.

*Light rays travel at 45 degree lines. Any massive particle must travel on a path whose tangent is always less than 45 degrees.

With that in mind, we can see there are 4 distinct regions. Let's start with the right diamond, labeled "Universe." This corresponds to an observer "outside" the black hole. A particle starts at the bottom of the diamond, in the infinite past, and eventually can reach the top of the diamond in the infinite future. That is, unless the particle crosses the horizon, and enters the top region, labeled Black Hole. Once an observer crosses the horizon, there is no way to get back out; one must hit the singularity (wave line) at $r=0$. These two regions correspond to a black hole that you have likely heard of.
However, essentially because of the time reversal symmetry you cite, there are additional regions in this spacetime. To the left, notice that there is a parallel universe, which can also throw things into the black hole. However, we can never cross from our Universe into the parallel universe, because doing so would require faster than light travel. However, the fact that there is a connection between these regions means this spacetime has an Einstein-Rosen bridge, or wormhole. In this case, it is a non-traversable wormhole, since no light or matter can make it from one side to the other. (Although, two observers from the two different universes could both jump into the black hole and meet there before hitting the singularity).
Finally, perhaps the most bizarre region is the bottom of the diagram, representing the white hole. Any matter that starts inside the white hole must eventually leave, and end up in one of the two Universes. If a black hole "eats" everything that falls into it, a white hole "vomits" everything that started inside.
The white hole is associated with a major theoretical issue, which is one reason why the maximally extended Schwarzschild solution is widely considered unphysical. Einstein's equations don't tell us what to do with a singularity; they equations break down there. It's possible that a quantum theory of gravity would tell us what happens near a singularity, but so far that is conjecture. For a black hole, this issue doesn't matter, because the singularity is hidden behind the horizon. From our perspective in the Universe, once something falls into the black hole, we know it hits the singularity and "something" happens, but we never need to deal with what the "something" is, because nothing can escape the horizon. (Except for Hawking radiation, but that's a whole other story). However, for a white hole, we have exactly the opposite situation. Is matter supposed to be coming out of the singularity? Is the singularity supposed to be doing nothing? We have no idea, but in order to solve the equations in the Universe part of the diagram, we would need to know what was happening in the white hole, including at the singularity. This makes the solution ill-defined.
However, a realistic, astrophysical black hole must form from the collapse of matter. (At least, most physicists don't believe there are "eternal" black holes that have always existed). The solution for a collapsing black hole is quite different (from here: https://jila.colorado.edu/~ajsh/talks/heraeus_ajsh_19/penrosecollapse_trad.html):

Here, the gray region represents a cloud of dust that collapsed due to its own gravity and eventually formed a black hole. As you can see, the "parallel Universe" and "white hole" parts of the diagram (as well as the anti-horizon) do not exist in this picture. Relatedly, the time reversal symmetry you pointed out is no longer present -- there was matter in the past that collapsed into a black hole; this fact breaks the time reversal symmetry.
Finally, it should be said that even this final diagram assumes a perfectly spherically symmetric collapse (it is called "Oppenheimer-Snyder collapse"). Because of all the symmetries involved in constructing these various solutions, physicists in the 1950s-60s or so widely thought that black holes were an artifact of assuming too much symmetry, and not a generic prediction of general relativity. (I take this sort of attitude to be the thrust behind your question -- you are in good company with some very good physicists in those days!).
What really started to convince people to take black holes seriously were the Penrose-Hawking singularity theorems, which showed that singularities do generically form in solutions of Einstein's equations, without needing to make any special symmetry assumptions. Penrose won the Nobel Prize for this work in 2020 (sharing it with observers who actually provided observational evidence for the existence of black holes).
The modern point of view is that the Schwarzschild solution (at least the "Universe" and "Black Hole" parts) are a good approximation to the equilibrium state of a non-spinning black hole after it has formed. This is also confirmed by numerical simulations.
