Everything you're saying makes sense, but I just can't get past what, in my view, is the absurd optics of, for example, an 8-9 team or possibly even 7-10 team making the playoffs. Maybe that example team had a really tough schedule, like you said, but if a team goes only 8-9 against a tough schedule, to me that says they're not worthy of making the playoffs. Now, it could also be true that an 11-6 team had a soft schedule and isn't really playoff worthy, but since they won a large percentage of their games, they deserve a shot to prove whether they're actually good. At least, that's how I see it. 
I remember the year the Patriots were 11-5 and missed the playoffs while the 8-8 Chargers won the AFC West and made it in. This board was all up in arms, but the Chargers beat the Patriots 30-10 that year. The Chargers also played the NFC South in which the worst record was 8-8 while the Patriots had the NFC West, whose best team was 9-7 and had 2 of the 4 worst teams in football in it. The Chargers ended up beating the 12-4 Colts in the 1st round before losing to eventual Super Bowl champion Steelers in the next round.
The schedule is so much more important in the NFL than any of the other sports because it is so short. Who you play and even when you play them just matters so much. That is why the NFL rewards division winners with automatic playoff berths and why they let those teams host playoff games.
That's a good example of what you're arguing, but I'm sure there are examples of the opposite as well, if I wanted to take the time to research, which I don't. Suffice it to say that you make a fair point, one I'll try to keep in mind when this year's South division winners clinch playoff spots with underwhelming records. 
Though I doubt my annoyance level will be any less, lol.
Teams under .500 making the playoffs
2020 - Washington - lost 1st game to eventual champion Tampa Bay 31-23 (the 2nd closest game Tampa played in the playoffs that year) on WC weekend
2014 - Carolina - beat 11-5 Arizona in 1st round, lost to defending champion Seattle in divisional round (Seattle would lose to New England in Superbowl)
2010 - Seattle - beat 11-5 New Orleans in 1st round, lost to Bears in 2nd round
That is it going back to 2002 when 4 division in each conference started. I suspect it didn't happen much, if at all, before the 4th division was added.
Over that same span there have been several 8-8 teams making the playoffs as well. In 2004 both the Vikings and Rams made it in as 8-8 wild card teams. They both went on the road in the wild card round and won road games, though both lost their 2nd road game in the divisional round. The Chargers team I mentioned also won its 1st game (and that was the same year that 9-7 Arizona made the Superbowl). The Giants made it at 8-8 in 2006 and they did lose the 1st round. The only other 8-8 team was Denver in 2011 and they also won the 1st game, before losing in their 2nd.
So since the advent of 4 divisions per conference, there have been 8 teams that have made it in the playoffs with a .500 or worse record and 6 of the 8 won their first playoff game.
