There are 375 and 416 Winchester 70s to be had out there lightly used for around 900 and new to near new for 1300-1500. Subtract those numbers from your 4000 cap and you have one heckuva piece of wood!
However there are a lot of factory Winchester 70s out there that already have really nice looking, correctly grained wood for strength in normal run factory stocks. Is this going to be a user for serious big and dangerous game? I've never seen burl or wood figure shoot anything. My thoughts would be to concentrate on a lightly used proven type rifle like a Winchester 70 and spend a relatively small amount of money on pillar bedding and maybe stock re-finish. The total $ for all that including rifle would leave a pretty good amount left over from the $2000-$4000 budget. Then add that surplus to the hunting trip fund.
Just thinking out loud.
Here's my lineup of DG rifles. All factory standard grade stocks. I pillar bedded all myself for only the cost of the alum pillars, bedding compound and my labor. I have left the BRNO as is. Seems their bedding is pretty stable and already a form of pillar bedding. Its group sizes and POIs are consistent, just as a properly bedded rifle has.
Top three- Winchester 70s
Bottom- BRNO ZKK
Not a dog in the bunch and even the most plain, the BRNO, is of good dense walnut and correctly grained for strength. Matter of fact the least attractive of the bunch shot a sweetheart of a group today- The BRNO 375 HH
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