So this is really the twist in the construction of QFT that is still sometimes misunderstood. Historically, people like Dirac noticed that solutions to relativistic field equations such as the spinor equation for $\psi(x)$ admitted apparent negative mass-energy densities. However, once you go into QFT, either from a "second quantization" or a Fockian bottom-...
When you time-reverse a black hole, you get a white hole. Therefore you don't see black holes as antimatter fountains. White holes would be antimatter fountains, but that doesn't say much because everything falls out of white holes anyway.
why isn't PT defined to be a reflection in (3+1)-dimensional space-time?
It is a matter of defining what is a reflection. In the following paper: About Chirality in Minkowski Spacetime, Symmetry, 2019, 11, 1320 (https://doi.org/10.3390/sym11101320), P, T, and PT, are all mirrors.