"But I would imagine a BH from stellar collapse becomes stationary after reaching a steady state condition?" Well, it all depends on how one models stellar collapse. One could use a spherically symmetric collapse of matter, where the metric inside the matter content is either say FRW type or Vaidya metric, but outside the collapsing matter, the metric is Schwarzschild eternally (by virtue of Birkhoff's theorem). The metric is eternal in the sense that if we consider the region outside the last shell of infalling matter, the geometry of this region will be given by Schwarzschild solution for eternity. However, one could also start from a generic Robinson-Trautman type metric, and using Lyapunov-functional argument it can be shown that for any general initial data on outgoing null surface $u=$const (where $u$ is the retarded time), the system will emit gravitational radiation and later settle down to schwarzschild solution (https://link.springer.com/article/10.1007/BF02431882). Likewise, one could come up with other models which is not eternally schwarzschild; it will depend on the symmetry and type of boundary conditions one imposes on the solution for Einstein's field equations. (EDIT: Even if the asymptotic state of solution is given by schwarzschild solution or any member of the Kerr family, it may not necessarily be a BH solution. A BH solution would also mean existence of event horizon and for schwarzschild case, it means that the matter field must be compressed within its schwarzschild radius. This will depend on the internal physics of the stellar matter which requires a separate consideration. For white dwarfs, the Chandrasekhar limit provides an estimate / upper bound of the total mass beyond which the star can collapse into a neutron star or BH, and in this sense the metric describes a schwarzschild BH (assuming spherical symmetry is maintained through out the process))
"Is it splitting hairs to say that non-spinning astronomical BHs (or stars and planets for that matter, outside their surfaces) are not truly Schwartzchild BHs, or is there a deeper difference?"
I guess by splitting hairs you are referring to the BH hairs. It was proposed by Hawking that highly energetic collapsing matter (not necessarily spherically symmetric) can introduce super-translations on the generator of event horizon (https://arxiv.org/abs/1509.01147). Since, supertranslation is an infinite dimensional abelian subgroup of the total asymptotic symmetry group (say the BMS - symmetry ), it has been interpreted that these super-translation effects on outer event horizon produces infinitely many soft hairs. One can also artificially implant such soft hairs on Schwarzschild horizon (such as https://link.springer.com/article/10.1007/JHEP05(2017)161). So, if a realistic static non-rotating BH needs to have such infinitely many soft hairs, it is no longer equivalent to the exact Schwarzschild solution.