I was wondering if a .308 or 30-30 would be a good choice.
YES! to the bolded one. Other perennial recommendations for this category (chambering-wise) are .243 Win, .257 Roberts, .250-3000 Savage, .260 Rem, 6.5x55mm, 7.62x39mm, .45 Colt, and .44 Magnum rifles.
But as mentioned, running with a .308, .270, .30-06 or similar with the Reduced Recoil ("Managed Recoil") loads is a good plan too.
Personally, I consider the 6.5x55mm round in a good turnbolt to be the perfect "deer rifle" to cover all conditions. But the .30-30 Win levergun is also definitely in the top 2 or 3 choices.
an effective range of "only" 225-250yds. is not really a handicap. Just ask any of the nation's 5 million black powder deer hunters and bowhunters.
While true, I would submit that the vast majority of muzzleloader/BP hunters' ACTUAL
effective range is around 100-125 yards less than their IMAGINED effective range of 250 as hyped by the makers of muzzleloading rifles, gear and scopes with ballistic reticle marks, purporting to enable hits for: "250 with 150 at 250 in a .50" - I.e. 250 grain sabot slug, .50 cal, 150 gr of BP equiv., at 250 yards. Mind you, effective not meaning what their gear is capable of, but what THEY are capable of as hunters under field conditions.
Regardless, as you say, not really a handicap since the overwhelming vast majority of deer shot with ANY weapon are taken at ranges under 100, and most of them under 75 yards.
The .30-30 Win is real good medicine, AND pretty flat-shooting,
at any/all ranges at which the vast majority of hunters can actually make hits. There's no ballistic disadvantage. The only disadvantage may be in the action. In very cold and/or very dirty environments, the turnbolt has the advantage of very positive extraction, in case you need a follow up shot.