There are/were several rifles under $500 that would shoot 1000 yds. The problem these days is the shooter, optics, steady rest and ammo and not the rifle. Though, if given the choice of 6mm Remington vs. 6.5 Creedmore to shoot 1000 yds, I’ll take the Creedmore. After 500 yds, the 6mm really starts to run out of gas.
The caliber is a 6mm Creedmoor, not Rem. It comes with a 7.7 twist and will shoot the 108’s etc quite well. They will easily go past 1000 yards, probably the majority of 1000 BR shooters are using various 6mm like the Dasher, etc.
I bought two of the Ruger Americans, one in .243, the other in .30-06, gave them to my son and oldest grandson. Both will shoot in the .5 MOA range, I think I paid $349 for the new one.
There are many rifles that can shoot at 1000 yard, here’s a few popular ones thats out there:
.300 Winchester Magnum
338 Lapua Magnum
408 CheyTac
50 BMG
These are calibers not rifles.