Very rarely do I think f5.6 is a deal breaker. I did have problems recent in the jungle but even a 300mm f/2.8 would be pushing iso 6400. The solution is a tripod, and then f5.6 is no problem again.
I think for tele lenses you need multiple lenses, a 70-300mm f/5.6 L ir 100-400 would make a nice light addition to the 120-300mm. If you need more reach then add a sigma or Tamron 150-600mm f6.3
I'm in Costa Rica now shooting humming birds, capuchin and howler monkeys, iguanas, agoutis, pelicans, owls, frigate birds and an amazing array of tropical birds, all with the Nikon 300mm f/4.0 and 1.4xTC o FF. Yes, sometimes f/4.0 is too slow, sometime 420mm f5.6 is still too short but getting that extra reach and extra stop is just impossible really. I Can't carry a 400mm f/2.8 around on such a trip even if I could afford it.
I mostly shoot around f7.1 with the TC just to get optimal sharpness and ensure sufficient DoF. Shooting wider apertures can actually cause issues, e.g you get a narrow part of the bird/animals body in focus and the head is blurred,