It's been a while since I looked at SteelChickens. They are always looking for the round that will reliably knock over a ram with the least recoil. 308 has NO problem knocking over rams. I lean towards the more recoil side of the spectrum. When a bullet pings a ram but it is still standing, I would have taken a little more recoil to put that ram under my belt.
The reason I write that is, you'll hear a lot of talk about specialty calibers but don't feel out-gunned with a 308. 10 - 20 years ago 308 was the most popular round, then it was 7mm-08. Now it is a toss up between 260 and 6.6X55. This trend was to reduce recoil, but ram reliability has also gone down a little.
One thing you can do to keep recoil down is use a lighter bullet for Chickens, pigs, and turkeys. These three don't put up much of a fight.
I forgot about your Tikka question. Again, about 85% of the rifles are built on a Rem 700 action. But I hear great things about Tikka's trigger, and the trigger is extremely important. I've never shot one, but I hear they can also be very accurate.
Another thing I notice about the silhouette crowd is free advice. They are not competing against YOU. They are competing against theirself. (at least in the lower levels) I'm sure when they get to master class they hold a few secrets back, but overall great people.