Please take in mind that the AP voting for its early years was EXTREMELY biased in favor of the big 10, mid-west schools, or USC. This bias probably shows up more in Tennessee records more than other schools however, because of the quality of teams UT fielded that went unrewarded for their success.
Also keep in mind the bowl games lost by national championship teams were in a lot of times played without many of the star players that helped make them champions to begin with, bowl games weren't terribly highly thought of except as exhibition games. That point goes for and against UT.
If you want to look at bias, the 1939 team is the flawless example. The last team to hold all opponents without a single score for the entire season, yet finishes number 2. This, in today's college football world, is an unthinkable feat. There is truly no excuse for this team not being awarded the national championship for the 1939 season, no one even scored on them.
If you want to be a purist, that's fine. But that assumes that AP was unbiased, using flawless voting systems, and that everyone actually paid attention to what anyone that wasn't in their conference was doing (which they didn't) and didn't vote based on their regional coverage preferences, which coincidentally was putting coin in their pocket. Count them all, or count none of them if you want, until there's a tournament it's subject to opinion.
Bottom line, UT had some very, very high quality teams historically that went unrecognized unfairly as national champions. In fact if you look it up there is probably fewer more championship worthy "bride's maids" for multiple occasions with national championships and Heisman trophies in the history of college football than UT. Tennnessee's football tradition and roots are far stronger than most know, even those fans of, or persons within the program. The team's history and traditions are stronger it's already impressive resume suggests, and recognized by biased polls or not UT fielded a hell of a lot of good football teams.