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Don't overthink the gun purchases. I've had several bolt-action rifles in my lifetime, and the basic difference comes down to where and how the safety operates. Yes, some (like my .303 Enfield) cock when the bolt is closed. My Browning A-Bolt cocks when the bolt is opened. Not a big difference when I'm shooting at a hog and working the bolt as fast as I reasonable can. The bolt release mechanisms are quite different, but when I'm cleaning the rifle I have lots of time to deal with that. The shooting part, aside from releasing the safety, is pretty similar.

I wouldn't worry about getting a .22 that's similar to your centerfire (and BTW I think .308 is a good choice — 'cuz I have that caliber also). Get a .22 that you'll enjoy shooting. I think the 10/22 is an excellent rifle — 'cuz I have one of those also. However, my usual .22 shooter is, as someone else already described, a Remington Nylon 66 that I bought when I was 16 years old (that was a looong time ago . . . )

The basic skills you want to develop are safety and accuracy. It doesn't really matter what rifle you use when you practice, just work on those things and you'll be fine. The differences between rifles is easily learned. Get the rifles you will enjoy shooting.