OP, you still didn't state what you want to do with the rifle.
Shoot targets from the covered bench off rests?? Hunt gophers, deer, moose???
What do you call precision shooting? Is sub moa good enough or do you want sub 1/4 moa groups?
Sub moa is easily available off the shelf but it requires a scope that is capable of reducing parallax to go with it and ammo it likes.
You asked a broad question and got broad answers.
If I may suggest, take a look on the CGN EE for a precision package that is already assembled, with quality components. There are usually several there at reasonable prices. Be careful about round count. If the person doesn't want to divulge that info, don't purchase. Worn throats aren't good for accuracy.
As for the Bergara???? It wouldn't be my first choice.
As for Savage, it also wouldn't be my first choice, but only because I think they're fugly. The nice thing about them though is that barrels are very easily swapped out and can be purchased pre chambered.
I've seen a couple of Savage rifles, with medium weight, 24 inch barrels, chambered for the 260 Remington perform on coyotes out to 500+ meters. The cross section on most coyotes is about 10 cm.
The two brothers that have those rifles love them. They picked the basic rifles up for around $400 each, new in the box, swapped out the barrels for $350 each and picked up knock off Zeiss scoped from Amazon, under $200. Both of the rifles already had laminated stocks and those were left as is, other than to relieve the barrel channels for better clearance.
Both of these rifles will shoot sub 1/2 moa groups out past 500 meters.
You can double or triple the cost by starting off with a Bergara as a base rifle. NO, I'm not putting down the Bergara. I like them, but if your budget is tight???