I was just bragging on the -06 in another thread.
If you are willing to handload, or get someone to do it for you, the 25-06 will amaze you. I use the same load for coyotes and groundhogs, a 75 grain Hornady hollow point. It is flat shooting, has intense range, and is more resistant to wind drift than the lighter .22 calibers. It has much greater long-range potential than the .243 (another caliber I like very much).
You also mentioned particular rifles. Anyone who reads my postings will recognize me as a devout fan of older Remingtons. Any of the older 700s chambered in 25-06 will make an excellent shooter. Mine will hold sub-MOA with no modifications, but mine is a 1974 vintage, and the newer 700s are not the rifles the older ones were.
My father-in-law has the Savage American Classic in 25.06. He bought it new last year, and it is a splendid rifle. We did do away with the Accu-trigger and replaced it with a Timney, but that was more a matter of personal taste, and the factory Accu-triggers have many fans.
Sadly, Thompson Center doesn't offer their amazing Icon in 25-06, or I would recommend it. My father-in-law bought his first Icon early last year, chambered in .243, and was so pleased with the rifle that he has since bought 3 more in .300 TC, .308, and .22-250. They are all Sub-MOA out of the box, and the quality is superior.
KR