Interesting subject. I have built 6 Savages now. I am just getting ready to complete a 6.5 SAUM as my latest build.
There is a budget rifle and then there is cheap. I prefer budget as I do believe you can do well enough with in your price range. 1 thing a builder has to keep in mind is reloading or factory ammo purchasing. All of my builds are for reloading and don't usually include the price of those components.
So, if you call me today and said you wanted a good accurate budget minded long range rifle scoped for $2000.00 I would say, yep, no issue.
Here are the price ranges I have found and built from. Keep in mind, you can buy used or new components. For savage you can order pre-fit barrels.
Stock- 200-300
Barrel- 200-700 used to new with brake. There is quite the swing here.
Action- 250-400
Metal- trigger guard, blind mag, bolts, lug, barrel nut (150)
Rings- 50-100
Picatiny rail - 70-100
Scope -800.00 Vortex Viper PST for example
You can take a cheap savage stock and make it work fine. I have a 6.5-284 in a standard stock that shoots .5 MOA, and several others that do just as well. I am not saying these are great stocks, I am just saying they seem to be fine for a budget minded build. You can bed them, strengthen if necessary etc. Or replace with Hogue, B&C, etc with a budget replacement. I took my accustock and modified with flush cups and am installing a custom adjustable cheek rest. I like this stock as it is light and strong.