You could go classic and run with the old NFL and AFL pre-merger (as Capanther does). Or, you could pick the 16 teams which have the most championship appearances in their respective histories for your league.
You could base your league on the old NFL 1938-1960 —the original teams plus the ones which lasted only a few years like the 1930s Brooklyn Dodgers, the old Dallas Texans, and the New York Titans to round out the lineup (and makes for an interesting twist in that you have the Cardinals in Chicago, the Rams and Browns in Cleveland, and a New York with three NFL teams).
Playing a season shouldn't take years. You could try using a system which one coach here uses to run his 120 team solitaire college league, in which he plays one possession set per side and the final score is determined by yardage gained in addition to any touchdowns or field goals scored. Pick one or two matchups as your "games of the week" to really indulge a full game. Playoffs and championships you play full games, of course. If you decide to go full games all the way, it should still be possible to play a 80/88-game season (10/11 games per team —which works out to 80/88 matchups) within one year's time. It will take a little longer, but would be the way to go if you're also going to keep player stats instead of a simple record of final scores.
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