My 6.5s are custom built, the 6.5-08 is a Winchester 70 short action and the 6.5 Rem BR is on a Remington 700 action. I have a 6.5 Rem Magnum on a single shot Rem 4

action but no real experience with it. They are all target rifles with long heavy barrels.
I don't hunt so I am not going to give advice about bullets (what weight or brand for different types of game) that you couldn't get out of a manufacturer's catalog.
I don't think there would be any difference between a 25 caliber rifle using a 117-120 grain bullet and a 6.5 using a 120-123 grain bullet if they were at the same velocity and of the same construction/type. For target use there are a lot more 6.5 target bullets available than 25 caliber target bullets. For varmint hunting there seem to be more 25 caliber bullets available, for big game (or larger than deer?) there are more 6.5 bullets available.
I had, for while, a Winchester 70 in .257 Roberts (made in the late 80s early 90s?) which I never got to shoot well. It would string vertically with the 3rd shot often 4-5 inches above the first and if I set the rifle the aside and shot something else for a while I could go back to it and get another vertical string. Glass bedding and recrowning helped but did not cure. But 1 rifle doesn't prove anything one way or another.