Indeed, one of the definitions of spontaneous symmetry breaking is in terms of its susceptibility:
Suppose we add a symmetry breaking perturbation $h \; \delta H$ to our
Hamiltonian (as you do), if
$$ \lim_{h \to 0} \lim_{N \to \infty} \langle m \rangle \neq 0 $$
then we say our system has spontaneous symmetry breaking.
(Note: $N$ is the number of spins in our system. Indeed, on a mathematical level, non-analyticities can only arise in the thermodynamic limit.)
What is special is that any arbitrarily small perturbation will do. Imagine you have a million spins. If the state is originally in a symmetric state (i.e. not symmetry broken yet), then even if I just apply an arbitrarily small magnetic field on a single spin, the whole system will choose that orientation.
You suggest that the fact one in principle needs the environment to 'make the choice' that this is not really spontaneous. It is true that in that philosophical sense of the word, the direction of magnetization is not 'spontaneous'. But what can be called spontaneous in the universe? If I perfectly balance an egg, then the direction it will eventually roll when it loses its balance is spontaneous (or not spontaneous) in exactly the same sense. And note that once the egg has rolled down (and stopped), the tiny perturbations in the air which influenced its original direction are now no longer sufficient to change its position. I.e.: after the `spontaneous' process, the system is now stable.
The same thing happens in the above magnet: once it has chosen a direction of magnetization, then changing the applied magnetic field on that single spin I mentioned before will not change the total magnetization. So in that sense it is not true that it is so susceptible! One needs to apply an extensive magnetic field (i.e. a field that acts on most of the spins) to change the direction of the magnetization.
That is what is so funny about these systems:
An arbitrarily small perturbation can create a magnetization, but it
cannot change it!
On a more quantum-mechanical note, if one has a Hamiltonian whose ground state should display spontaneous symmetry breaking, then if one takes the ground state to be in a symmetric superposition (which one can always do), then this state has ridiculously long entanglement. These are called cat states (in reference to Schrodinger's cat). This is a natural consequence of the above: an interaction with a single spin has to influence all spins at once, which is only possible if every single spin is entangled with every other spin. An example is the state $|\uparrow \uparrow \uparrow \cdots \rangle + |\downarrow \downarrow \downarrow \cdots \rangle$. (Indeed: an interaction with a single spin will collapse this 'cat state' to a product state, and then it is clear that any subsequent single-spin interaction cannot flip the state to the other product state.) Indeed, the way symmetry breaking phases are classified in one spatial dimension is in terms of these entanglement properties [Schuch et al., 2010].
EDIT: the short answer to this question is that a time-reversed black hole is a white hole, full stop, so if you apply time-reversal to a particle falling into a black hole, you get a particle falling out of a white hole, but we don't physically expect to observe white holes.
Original text:
A blackhole space-time does not violate T-symmetry because, the extended Kruskal solution also contains a white hole:
https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates
so, if you time-reverse the portion of a curve falling into the black hole, it will become a portion of a curve falling out of the white hole.
Now, we expect that the universe was created with initial conditions that don't allow white holes to exist, but this would mean that the T-symmetry in GR is spontaneously broken by some quantum theory that is not GR. It is absolutely present in the schwarzschild and Kerr spacetimes, though, thanks to the extend kruskal coordinates trick.
Best Answer
You missed something: the gravitational waves.
A black hole merger spacetime contains gravitational waves leaving the merger at the speed of light. Time reversal reverses time across the entire spacetime, and this converts those escaping gravitational waves into a converging gravitational wave front, as well as the black holes into white holes. These waves converge in on the central (now-)white hole and get so strong at that central point of convergence as to be able to "buck" it apart into two separate white holes.
Without those incoming gravitational waves, such a split would not occur.
EDIT: As A.V.S. points out in the comments, in fact, a better answer to this question would be that the future evolution of white holes is in general undetermined, or better unrestricted, in the sense that multiple future trajectories from identical phase-space points will satisfy the dynamical equations, though of course that means still that we must highlight that a crucial element of the answer here is that the time reversal turns the black hole into a white hole. (Indeed, this is part of why they're called "white" - technically that's understating it: they can literally spit out anything - even unicorns, no seriously, it'd be entirely [though unlikely] consistent with the equations for a 1-horned ungulate to pop out, as much as literally anything else.)
In a realistic black hole collision case, which is what I assumed in the answer above, then of course, yes, you will have the gravitational waves and so forth and you do have to take them into account in the reversal. But the situation is even more serious.
Since the future evolution of white hole is unrestricted, you can build scenarios with a totally causeless, spontaneous split of the white hole, and have it be consistent with the dynamical equations. As it is a consistent evolution, it doesn't violate time reversal symmetry. The reason that the Universe isn't covered with tiny white holes is that they are next to impossible to form in the first place - and likely, general relativity is not the final description of these things.
(I want to point out that there is actually an analogy for this within ordinary Newtonian mechanics called "Norton's dome". It is not physically achievable, but is still a system within the mathematical theory which has a similar property of its present state being equiconsistent with multiple future evolution trajectories.)