You are right, the Rifle shouldn't be, that "picky" !!! I looked at my box of, .270 cal. 140 grain Bal Tip's and Nosler shows that, they should expand, down to 1,900 FPS, so that is 500 Yards, for your 7mm-08 load. There are still some things that, we don't quite know about, Rifle Barrels like, why one bullet shoots "well" and another, "stinks" and yet, some guns may shoot, several different Bullets, "well"! There are many things, "in play" when it comes to accurate, Rifle barrels like,.. twist rates, velocity, bore size, land with, throat/ bullet length, powder burn rate and the list goes on. There probably is, a real accurate, load/ bullet for your Rifle but, you'd probably have to hand load or, test a lot of Factory Ammo ! If the Win 140 gr. Sil Tip shoots, about a 1 MOA group, say a 5-6 inch Bench group at, 500 that would be good enough for Deer/ Elk hunting. Your group will open up a bit, using Field position methods (sitting, shooting over "Sticks" or, Prone with a Bipod / over a Pack). Once you shoot a lot of practice, at Hunting positions and different Yardages, you should have, more confidence in your Rifle and Load. Good luck !
PS; do you have a Bubble Level, on your Scope? What power Scope, are you using ?