Are you a reloader?
No, but I'm willing to learn.
Well, that narrows things quite a bit. I will answer based on the fact that you don't reload & if you reload later, you will also buy more rifles, trust me on that one & at that point this question won't matter anyway.
1. Based on this, forget the oddball stuff with hard to find & expensive ammo.
2. Deer is the biggest animal, so no need for rounds of over 30 cal, they won't give you anything for ANY application stated over a 308.
3. You mentioned the desire for a round that could handle a 300 yard shot & I would rule out the 30-30, 35 Rem & such.
4. The 308 family rounds, especially 260, 7-08 & 308 will fill your needs. As far as noise is concerned, you would not tell the difference with any of these. The 260 & 7-08 have great long range performance, but the edge over a 308 comes into play after 500 yards, so for your use that would have no relevance.
Several rounds could be pressed into service, but the clear choice for me is the 308 for your needs. Out to your maximum distance of 300 yards it is extremely accurate. Many, as Barness states over & over, the 308 is about as accurate as it gets for a hunting round. The recoil is decent. The ammo is very common & economical (relative to the others), you can even plink with Mil. stuff since you don't reload. If you decide to hunt Elk, it will do that at 300 yards too.
It would be an easy choice for me.