I’ll be the odd man out and suggest, what has been the stereotypical all around western elk/deer/antelope rifle since the 1970’s - a 7 Rem mag or PRC. If this is to be the only big game rifle, it really needs to be paired with a 243 one way or another - nobody wants to shoot a 7 mag as a plinking or training rifle.
The weak link in a hunting rifle is the scope, so with a $3k overall budget, spend half that on the scope/mount. There are many flavors of rifle in the $1k range - there’s nothing wrong with Tikka, or Savage, Christensen, Howa, Mosberg, Ruger, Winchester or Remington. Get a replacement trigger whatever gun is chosen.
If someone doesn’t already have a 223 or 243 to practice with, I’d still go with $1.5k for the scope/mount and get a used 7 mag paired with a used 243, sharing the scope between them. That combination is ideal for all ranges and covers all animals found in the lower 48 from prairie dogs to elk. I’d much rather have the two basic rifles with basic stocks, than one fancy gun with carbon fiber stock that recoils enough the shooter doesn’t enjoy plinking with it.