Actually a lot of guns could use "tinkering." I believe bolt actions theoretically should be more accurate than a break open gun but I get pretty decent accuracy from Encores and Contenders (from rifles 1.5 MOA at the worst and generally in the range of 0.75 to 1 MOA). If you'll be happy with nothing more than 0.25 to 0.5 MOA, then I'd recommend a bolt action (but you'll probably need to tinker to get that accuracy in an off the shelf bolt action rifle). My experience with the T/C line includes 17 HMR, 22 LR, 22 WMR, 357 Magnum, 357 Herrett, 44 Rem. Mag., 222 Rem., 223 Rem., 22/250, 270 Win., 7mm/08, 7mm Rem. Mag., 308 Win., 30/06, 30/30 Win., 375 JDJ, 375 H&H Mag., 454 Cassull and 460 S&W Magnum. It's obvious I need some more like a 243, a 25/06, a 416 Rigby...
I agree, the Encore trigger as it comes from the factory is too heavy but that is easily remedied by a trigger job.
I've found T/C customer support to be no excuses and the best.
As for calibers, for 300 yard Montana shots I would probably go with the 7mm Rem. Mag. It will work for close shots as well, ammunition/components are of excellent selection, it has good inherent accuracy, etc. The recoil is also very acceptable. I have an Encore 7mm Rem. Mag. barrel with a brake (I ordinarily don't get braked barrels but the price was right) that is a very sweet shooter. It's one of the uncommon barrels that shoots the same place clean or fouled, hot or cold. I get about 1 MOA or less with Barnes 140 gr. XLC XBT bullets and Hornady 152 gr. A-Max bullets and the recoil is such that a wuss could take it.