#1 tip on building on a budget--make friends with a retired machinist who can do threading/barreling for you! Otherwise get a Savage where you can put on your own. You could be lucky enough to find new, armory-stored original FN or Mauser barrels. Used to be easier to find those, I know.
Absolutely! There are plenty of Remclones out there that, in theory, take prefit barrels with no machining work required. In practice, I'm four "prefit" builds deep now, and two of those barrels needed some lathe time. This was two different barrel manufacturers and one action manufacturer. In one case, the barrel manufacturer missed the undercut on the tenon threads so the barrel didn't bottom out on the face of the receiver. In the other case, the barrel had excess headspace. Both were easily fixed by my local gunsmith, but if I didn't have a good local gunsmith I'd have been shipping the barrel and action to get them sorted out.
Barrel nut builds aren't limited to Savage. There are barrel nut options out there for Remington, Howa, and maybe some others as well. Given the option, I'll always go with a shouldered barrel, but again, I have a gunsmith just down the road to handle any necessary machine work. If I didn't, the barrel nut setups would have a lot more appeal to me.
Having your own shop to do the rest of the work also helps cut down on cost. For the upcoming M70 builds I plan to do all the stock work and bluing myself. The rest of the metal work will be outsourced to the gunsmith since unfortunately I don't have a machine shop in my garage and my employer would frown on me building rifles at work. For a basic build, namely a refinished factory stock, no sights, and rust bluing it myself, I would expect to be just under $2k for the M70 build. Having a retired machinist buddy who will work for beer would certainly knock a few hundred off the cost. Realistically, I'm going to be buying more tools and adding express sights, so the cost will increase accordingly.
I'll be shocked if the .458 comes in much under $4k since I plan to go a little wild with a quarter rib, etc. The second M70, whatever it ends up being, will cost less, both since it will be more basic and since I'll already have all the tools.
It's a total rabbit hole, but it's fun.