I have a 15 month old so I think I can answer some of this. Her first photo was taken in the delivery room with my E-PM2 (by one of the nurses). That camera has also taken most of her photos but I also have a Nikon D5100. The Nikon has a 17-70 (so a 24-105 FF equivalent) 2.8-4 zoom on it which at times makes life much easier. I prefer primes (as much new baby photographing is indoors) but sometimes a zoom just makes things easier in tight spaces. My Olympus 45 1.8 takes the best portraits.
So, do you have a flash you can bounce off the ceiling? That's probably the most useful for freezing motion. You're not going to want to be flash crazy when he/she is brand new but as they get bigger, its' really helpful in getting photos which aren't blurry. That would be my first investment. IS helps if you're moving (shaking hands). It does nothing if the baby is moving though. You need a faster shutter (more light) or a flash to freeze motion.
In terms of video I do occasionally use the Olympus or Nikon (or even my iPhone). However the camera used most for video is a GoPro Hero 4 session. It's waterproof / crush proof and I can put it on the floor next to her. If she picks it up and throws it or tries to chew on it, it doesn't matter. We had a Hero for a bit, but the session has better audio since it doesn't have the lock in plastic case. This was also helpful for her first time in the pool photos

. I got the suction cup mount which I put on her pushcar/wagon and we do timelapses of her and me driving around the neighborhood.
Right now my challenge is that she's figured out walking at 75 mph and if she sees me on the floor with the Olympus, she charges me, sits in my lap and plays with the pilot button on my flash (pew pew). Which is really cute but doesn't leave me with photos. The Nikon is going to get pulled out again as the Olympus just can't focus fast enough. But that won't be an issue with a brand new baby.
I think your A6000 is fine (I wish my mirrorless had hybrid autofocus). Get started and see what you feel you need to get the photos/video of your child you want. I'd consider getting a bounce-able flash if you don't have one and get good at taking those.
Any for everyone joking about 5-6 years - my 15 m/o daughter works the Pen Mini fine. Touch screen, click on what you want the photo of. Boom.

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