As I’ve stated many times. I bought a Tikka T3x Hunter in 7-08 last year. It’s a little more than the budget you mentioned but it’s truly a great rifle. Very accurate, smooth action, great trigger, perfect weight and feel for me. Also it has a nice classic looking wood stock. To be honest I’ve only shot 2 types of ammo through it. Hand loaded 120gr Barnes TTTSX and the 140gr. Accubonds. Shot both rounds extremely well but the Accubonds shot just a little better. So far everything I’ve shot from whitetails, Axis, coyotes, and big hogs have not made it more than 60yds. My buddy has a new Browning BAR in 7-08 and he shoots the 150gr Norma Whitetail they shoot and perform great. I would think and of the Major rifle companies produce rifles that shoot better than most of us can shoot them.
Lots of people come to use my range and I’ve seen everything Mossberg to high end Brownings. With the right round they all shot great. I’m partial to Tikka and the higher end Savages. Ruger American is a great shooter but I’ve noticed some ejection and feeding issues with some.

It's hard to kiss the lips at night that chews your a$$ all day long.