This isn't just about antibiotics, it's also about monoculture. Both of those combined lead to a never ending arms race.
If the same kinds of AMPs appear across mammals and birds, breaking the monoculture (by e.g. raising animals of different breeds and species etc) but still using this class of antibiotics would still seem likely to cause the immune resistance problem. I'm not advocating for the monoculture-based practices, but I don't see how it's related to this specific mechanism.
We see a similar result happening where our economic system forces us to depend on monocultures for food production.
Did you mean, "the great filter?"