If the intention is to shoot big game beyond 300 yards I concur, but coyotes and such are good targets for a .308 out as far as you can hit them. Given a 165 Sierra at 2500 (probably chosen for accuracy rather than effectiveness) the bullet has slowed to 1900 fps at 300, and IMHO this is below the velocity where the bullet will function reliably as designed on a big game animal. There is more to the big game bullet than the accuracy required to hit the target. At impact the bullet must have the velocity and the appropriate construction to deliver a lethal blow. It can deliver that blow in a number of ways depending on the design of the bullet, but there must be sufficient velocity to allow that to happen and create a deep and wide wound channel, even when big bones are hit. Bullets that are designed to function with impact velocities up to 3000 fps on big game, tend not to function as reliably one the velocity has dropped below 2000 and almost without exception at velocities below 1700 they will pencil through. In such cases pointed bullets will cause minimal wounding, and almost no blood trail, although flat nosed bullets do better, sometimes much better. Choosing a .300 Winchester significantly extends the range of a 2000+ fps impact compared to the .308.
I can't imagine how you found a .300 Winchester more tolerable to shoot than a .270, but perhaps the .300 fit you just right or perhaps you were feeling sensitive on the day you shot the .270. Anyway, if reach out and touch stuff at long range is your ambition, the .300 Winchester is sort of the entry level to the world or real recoil. It is unlikely that would would be able to shoot as many rounds per sitting with a .300 as you can with your .308, before your trigger control began to slip. Rocky Chandler wrote a very good piece praising the attributes of the 7 Mag for long range work. While some of those advantages don't have significance for a civilian shooter, the combination of long range accuracy and flat trajectory with moderate recoil and blast is hard to ignore.