So, they and Chicago did the right thing by withdrawing from their former leagues, as I stated. As for your last statement, it pretty much depends on whether Tulane ever gained any competitive traction.
Tech had considerable football heritage, but by electing for the Bobby Dodd football-is-fun template, they knew they wouldn't be able to hang with the helmet-busting game that Bear Bryant brought to the league and that others assimilated.
Back then, in football, each school could pretty much select which 6 of the 11 other conference members they would play each season. Which is how Georgia Tech could get away with not scheduling Ole Miss State at all, i.e., during Tech's 30 year run in the SEC they did not play the Maroons on the gridiron
at all. And they played Ole Miss only twice - a regular-season game in Atlanta in 1946, and in the 1953 Sugar Bowl (in that era, the Sugar Bowl mostly matched up SEC members on Jan 1 in order to keep things lily-white).
I have a feeling that if Dodd had been willing to schedule the Mississippi schools even occasionally on the gridiron, they may have been more empathetic to his concerns regarding the 140 rule and sided with him. But Dodd was also looking at making Tech more of a national brand than a regional one, and was kind of looking to leave the SEC anyway. And in 1963, Yellow Jacket football was
the game in Atlanta. We know how well it turned out 15 years later. I guess it never occurred to Dodd that the NFL might place a franchise in Atlanta at some point. I disagree with your assessment of Tech football - they had well-sustained success during their years in the SEC.
And Tulane was competing well enough in the SEC in sports not named football. I think that football took a downward turn after WWII. Eventually, they had asked the SEC for football "relief" which was rejected by Birmingham. Not sure exactly what the Green Wave had in mind, but this began a process of de-emphasizing athletics. That is, until many alumni realized was was happening, and "stabilized" Tulane as a competent enough mid-major. Today, they regularly schedule LSU in all sports except football.
I'm certain that the floundering of Tech and Tulane post-SEC stifled any possible thoughts in Nashville of departing as well (if indeed there were any). I think of Tulane and Vandy as having similar demographics and aspirations. But one school stayed, and one school left.