As far as GT's claim, undefeated seasons weren't enough because they were fairly common back in the day. Not as many games and big discrepancies in talent.
There was only one year in the 1950's when there weren't multiple undefeated teams at the top of the polls (1959). Some years there are several. This is why we have the CFP, which I for one think has been a huge success. In the five seasons of the CFP, the winner had a regular-season-ending CFP ranking of 4, 2, 2, 4, 2. As long as we avoid rematches, I think it proves the point of having a playoff.
What's interesting, though, is that in the first 3 of those 5 years, there was a single team to finish the regular-season undefeated – and hence they would probably have had a strong claim to the crown in pre-playoff days, but none of them went on to win it (FSU in 2014, Clemson in 2015, Alabama in 2016). That seems kinda crappy for them. They won more than teams in earlier eras won, but it still wasn't quite enough. (This year obviously there were three undefeated's, which is what the CFP was built for.)