My argument had several scenarios and you merely picked the one that was the easiest to attack. Regardless, let's focus on the key point, why 14 teams is better than either 12 or 16. The simple answer is 25 years of statistics. It's much more common that the 7th seed has a positive record than the sixth having an even or negative record, so the likelyhood that a seventh playoff spot would lead to many 8-8 teams making the cut isn't big. Additionally, the entire idea to have 14 playoff teams exists because it is so common that the 7th seed has a positive record. However, if you look at the 8th seed of any given year, the average record is close to 8-8 and that's why 16 playoff teams is too many.
I wouldn't say that you being a Patriots fan works as support for your stance, because it can be easily viewed the exact opposite way. Because your team didn't get into the playoffs with an 11-5 record (something that has a less than 1% chance of happening), no 7th seed shall make the cut, ever.
Finally, there's the argument that a 7th playoff spot in each conference would give more teams something to play for towards the end of the season, and that's something you keep ignoring. You are so focused on that one additional playoff game per conference that might suck that you lose sight of those many meaningless games that are played each December. The NFL tried to soften that problem by making week 17 all about divisional games in the last two or three years, but that didn't really help much. Despite bitter rivalries, many teams opt to phone it in. Of course, meaningless games will always exist, but 14 playoff spots would reduce the number of such games more than anything else the NFL has tried.
Firstly, I picked the scenario that is most likely to happen under a system where the teams with the six best records make the playoffs, as I am proposing. If we implemented a seven team scenario, with the top 7 records making the playoffs, last year would have included the 9-7 Houston Texans and the 7-8-1 Carolina Panthers (who admittedly did make it under the equally silly current playoff rules). The year before, the 8-8 New York Jets and the 8-7-1 Green Bay Packers (again, admittedly made under the silly playoff rules) would have made it. You have to go all the way back to 2012 to find a single team better than 9-7 who would have benefitted from this, and even there, it would have resulted in another .500 or worse team (the Pittsburgh Steelers) making the playoffs in the other conference. The VAST majority of the time, the seventh best team is a fairly mediocre squad as opposed to a genuinely good one left out because the conference was so stacked.
I can't say I really agree with the exciting point, either, especially now that the games are divisional, as you mentioned. So instead of two teams phoning it again, we get to watch a couple of really desperate mediocre teams playing against good teams that have either already clincehd their division and are playing their backups or against bad teams that won't care regardless. Either way, it's not particularly exciting.
More than anything else, what I dislike is punishing the number two seed by taking away their bye just for the sake of letting what is usually a pretty mediocre squad in the playoffs. If we have to operate under division winners get into the playoffs regardless of how bad they are rules, then I'm more open to it, but nothing about this strikes me as better than a system where the top 6 records get in.