Yeah, the only big problem with approx. sqrt is that it's not consistent across systems, for example Intel and AMD implement RSQRT differently... Fine for graphics, but if you need consistency, that messes things up.
Wait, what? Do you have a resource I could read up on about that? That is moderately concerning if your math isn't portable across chips.
Newer rsqrt approximations (ARM NEON and SVE, and the AVX512F approximations on x86) make the behavior architectural so this is somewhat less of a problem (it still varies between _architectures_, however).