Remington initially gave the 244 a 1 in 12" ROT and offered 2 different 90 grain factory loads that were intended for shooting medium game, a soft point and a bronze point. My 244 is a late 725 that has a 1-10" ROT, so no problem loading 100 grain bullets. If you want to hunt with the 722, I'd recommend trying the 85 grain Partition, 90 grain AB, and/or 90 grain BT. Bullet placement and construction counts for more than 10 or 15 grains of bullet weight, or so it seems to me.
I'd expect the Ruger 77 in 6mm REM to have either a 1-9" or 1-10" ROT, so 100 grain bullets would work fine.
AFAIK, the only cataloged 250-3000 bolt action rifles made in the past 50 years that have a 1-10" ROT are the Rugers and the Remington 700 Classic. The Winchester 70 Lightweight Carbine has a 1-14" ROT, a real head scratcher for a rifle cataloged when the only cataloged factory loads had 100 grain bullets.
I don't think that there is any practical/meaningful difference between the 6mm REM and the 250-3000 if you're shooting comparable bullets. The 6mm REM has a little more case capacity than the 250-3000, so you could get a little more velocity out of it, but not enough to be significant in most cases.
