I have an old (very old) Sako Finn Scout that I bought about 12 years ago second hand from a professional gamekeeper I knew. Having been used by him over a number of years for virtually all of his (extensive) pest control shooting it had seen a great deal of use when I bought it. I know that I have personally put the best part of 40,000 rounds through it myself over the years, and it still shoots better than any of my friends rifles. Groups are in the range of 0.5 / 0.6" at 100 yds with Winchester 40gr subsonics (excellent rabbit ammo) and I've never had a problem with rifle at all. If build quality, finish, reliabilty and accuracy are as good on the more recently made Sakos as on my old thing, I would not hesitate to recommend a Sako to anyone.
I do also own a stainless 10/22. I have fitted it with Volquartsen trigger parts and extractor and stuck it into a Hogue overmoulded stock. It's now a very comfortable rifle to use and I have no feed problems with it at all as long as it's kept relatively well cleaned. Again, I use the Winchester subsonics in it for bunnies. The 10/22 isn't the "tack-driver" that my Sako is, but it's still more accurate than I am.
I have never owned a CZ / BRNO but judging by those that I know friends have, they are good solid reliable rifles that warrant the money you will pay for them.
Lets face it, just about any modern production rifle is going to be more accurate than the guy holding it. What matters is that you are happy with the rifle, that you are comfortable with your set-up and have practiced enough to ensure clean one-shot kills on anything that you are ever going to point the thing at. For rabbits, that means being able to put the bullet in the head EVERY time. The shots that you take whilst out rabbiting are nothing like those you are likely to take on a rifle range. Shots will taken from odd positions, free-standing, rested agains trees and fenceposts etc etc, so these are the sort of shots that you need to practiced at, not just comfortably rested or prone shots.
Keep safe and enjoy yourself !