I think 400 or 600mm FF equivalent, ideally 600mm, primes are fine as you are almost never too close. The trade-off is that the longer the focal length, the more difficult it is to get the bird centered in the frame. Also, with the A1+600mm you have 45MP vs 20MP for the OM-1+300mm. In the OM-1 option I doubt between the 300mm f4 and PL 20mm f2.8.The framing of the Olympus combo is the same as the Sony combo.
You could also look at the 100-400 Olympus lens instead of the 300 but maybe that would be too slow an aperture? What FF equivalent FOV would be your ideal to frame with? There are some other m43 options with shorter focal lengths and f/2 or f/2.8 apertures that maybe would work better? Like a 200mm that would frame at 400 FF?
From what I've read on this Olympus OM-1, for short bursts with quick acquisition it may be the best out there but for longer tracking sequences the A1 will best it in tenacious tracking.
The A1 tracking is very good indeed, difficult to assess how much behind the OM-1 would be, I got some nice shots because of the great tracking... also missed some because not acquiring focus fast enough. If they were worth the same, I would pick the A1 I think, but I have not tested the OM-1...
Would the 500PF f5.6 be too slow? I am also thinking about that, but I think there are a few happy users here of that lens for BIF, so hopefully they will be able to shed some light. In the end the D850 is not miles and miles behind (for me) and you have a good cushion of MP if you need to crop (as with the A1).