WW,
I believe Burns builds his own stuff, most I have seen were Rem 700 based ,McMillan stocked, Leupold's modified by premier reticle etc. Those are by sight observations not confirmed. There are several people that are very capable LR gunsmith's myself included, Kirby Allen also posts here does some very fine work, Chris Mathews comes highly recommended. If I was making a recommendation to a customer with a request like yours I would suggest a Remington 700 platform (they are the small block Chevy of the gun world), there are several setups that meet or exceed the performance of the rifle Burns used in his videos. The best thing to do is to set down spend a few minutes on the phone with an LR smith and figure out exactly was your wants and needs are, then get a package that best bridges the $/performance/caliber/weight/optics/and technique you want to use. You will of course find many opinions of what is the best barrel or stock, etc. I have my own preferences for those items and base it on having built many LR rifles not just hearing that a particulr product is the best. Everyone will have their opinions. Just be sure who ever you choose has previous LR rifle experience and gives you confidence when you talk with them.
Shawn