Thank you for a well written response. Unlike many others you admit that there is some cloudiness to how you rate a team. The championship (or lack of it) is what actually clouds the idea of who was "better" in these cases. Different sport example, the 85 and 87 Celtics were better than the 81 Celtics. (Or the 79 Sonics or Bullets, whichever team was the winner)
The reason the ELO ranking systems does not care at all about who won the Super Bowl in determining who is "better" or even "best ever" is mostly due to single elimination. It takes into consideration, when it comes to wins and losses, just the record. And the 2007 Pats were 18-1, the best patriots team of all time.
The other thing it does is crunch numbers. The Patriots in 2007 usually destroyed all comers, most of their opponents. They had the number 1 ranked offense and like the 4th ranked defense. The won by an average margin of victory of 20PPG, which is astounding. The ELO ranking system has the 2007 Pats as number one "BEST" team of all time. It ranks the Super Bowl Champion 2004 Pats as the 4th Greatest team of all time.
I get what you are saying about GOAT being need the trophy, and believe me I LIVE for the Patriots winning the Super Bowl. But winning that one game does NOT (and this is a fact) determine which team is truly BETTER. It proves who played better and was better on that one game and day. That is one of the great things about football actually.
If the 2001 Pats (ranked number 338 ALL TIME by ELO) had played the 2001 Rams 10 weeks in a row what would the 2 teams records have been against each other? That is part of why makes football so exciting. BUT that one game, does not determine the better team, some of the time. (Obviously a lot of the time, the better team does win.)