An edited picture from tests out and about today. I'm still on the fence, I was expecting a better hit rate with the A9 and more accuracy on the eyes; results today were still around the 50/60% no matter the differing settings I used. Would an OM-1 (and the 12-40 Pro or 40-150 f2.8) produce a better hit rate and more animal eye AF attributes?
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Well, it depends (unfortunately). I haven't used the Sony a9 so I can't speak to it's performance so I am coming from Sony a7 Mark IV (which does 20 AF calculations per second instead of 60 like a9 but it has newer processor and AI algorithms, for whatever that's worth).
With Sony a7 Mark IV I had 8 out of 10 pin sharp focus wether it was Sony FE 85mm f 1.8 or Tamron 70-180mm f 2.8 (Mark I) but I had very few choices in body positions (like floppy up ears, or tongue out, or front paws up, or all paws off the ground, etc) . Which made me shoot 3 to 5 bursts with my dog and in the summer she would get tired quickly, in the winter she would lose interest in running and playing after one hour (instead of 2 or 3).
With Olympus OM-1 I get many more shots of the body position I want and decent in focus (about 70% ... So if I shoot 25 FPS for 5 seconds I get 125 images of which 87 images are sharp and that's a higher chance of getting the image I want). At 50 FPS you can get even more options but I wouldn't use that unless it's an extremely fast dog (like jumping Border Collie and dog sports competitions).
As for pin sharp Eye Tracking I don't think Olympus OM-1 is better or the same as Sony. To be honest the images you shared so far are quite acceptable for me (especially for personal use) and even for professional use.
One thing to make sure you know first is that some lenses on Sony will be able to give you a look/rendering that no Micro Four Thirds lens will give you ... Especially for similar price and performance.
For example the fastest telephoto lenses you can get are Olympus 45mm f 1.2 PRO and Panasonic Leica 200mm f 2.8 that have outstanding AF performance (which you would want to keep up with a fast dog at 25 and 50 FPS). Though Panasonic lense are capped at 25 FPS max.
The Olympus 40-150mm f 2.8 PRO can give you similar DoF and look at f 2.8 and 150mm to the Sony FE 85mm f 1.8 wide open but you are at 3 times the distance for similar result. The Autofocus keeps up well at 25 FPS in almost all circumstances and at 50 FPS in better and best light.
I wish there was a simpler answer but, unfortunately and the truth is that, you need to look at a system holistically to guage the performance you need. I'm saying this out of painful learnings from switching fully from Sony to Olympus last year.
I wanted to trade IQ for performance and features that (only at an affordable price) Olympus can give: Pro Capture, 25 and 50 FPS with C-AF and Blackout Free, Animal and Bird Subject Detection for shooting dogs in action (sports or playing).
I have yet to test the Olympus 45mm f 1.2 PRO but that's my first priority next year. I have tested the OM-1 on (depending on the light available):
*Olympus 12-40mm f 2.8 PRO: Excellent
*Olympus 40-150mm f 2.8 PRO: Exceptional
*Olympus 100-400mm f 5-6.3: Average to Bellow Average
*Olympus 45mm f 1.8: Excellent to Average
*Olympus 60mm f 2.8 Macro: Good (full AF Limiter Range) to Excellent (Far AF Limiter on)
*Sigma 56mm f 1.4 DC DN Contemporary: Average to Bellow Average