Has a team ever entered the last game the 1-seed, lost and missed the playoffs?
If you are a 1-seed and lose a game you’ll almost certainly get a Wild Card spot under the current scheme. In order for a single game to knock a 1-seed from the playoffs then the top 6 teams in the conference must be within 1 game of each other and I don’t think that has ever happened.
I think I found one… In 1979, there were 5 playoff spots per conference (3 division winners and 2 wild cards) and after week 16, these were the standings in the NFC:
NFC East
Washington: 10-5
Philadelphia: 10-5
Dallas: 10-5
NFC Central
Chicago: 9-6
Tampa Bay: 9-6
NFC West
Los Angeles: 9-6
At that time Washington was 2-1 against Philly/Dallas, Philly was 2-2 against Washington/Dallas, and Dallas was 1-2 against Washington/Philly - so Washington would have been the #1 seed.
As it turns out, Washington lost to Dallas in week 17, Philly beat Houston, and Washington missed the playoffs altogether.
Those were weeks 15 and 16; the NFL didn’t have bye weeks before ~1990.
Otherwise, that sounds like a feasible example, esp. as Chicago and Tampa Bay both won in week 16, as well, giving the NFC a clump of five teams at either 11-5 (Philadelphia, Dallas) or 10-6 (Tampa Bay, Chicago, and Washington). Dallas won the NFC East at 11-5, and Tampa Bay won the Central at 10-6. Philadelphia was the first wild-card seed (also at 11-5), and Chicago got the other wild-card seed.
But IIRC, there were not true seedings in '79,
Wild card teams played on opening week, but could not play the champ from their division until conference championships.
It was early 90’s before a true seeding playoff was instituted