The 7-08 has only become somewhat popular the last couple of years here in Texas. I have shot 2 different ones, a browning, I belive it was a micro medalion, and that sucka kicked like a mule, the other a rem 700 sps I believe, hardly any recoil.I am sure the 7-08 is a great deer rifle but a bit excessive for varmints, I think the .243 is excessive for varmints as well. To me the .243 is a deer rifle, never thought of it as a varmint rig.
The only problem with the 7-08 around here is shells, you can find the rem 140 gr corelocks pretty easily but thats it, no selection to try different ammo, might be different where your at. I would go to the stores and see what you can find on the shelves first then make your decision. But for a deer and varmint rig the .243 would be hard to beat, maybe a .223 would be a choice also.