Ive really been meaning to come back and do an answer for you. And having now filled out my brackets for both I can finally motivate myself to sit down and write, so I apologize for the delay!

And the answer mostly boils down to 3 things. NCAAT selection limits, weakness of the NCAA in capitalizing on their built in advantages, and the benefits of playing in the NIT for media exposure pre ESPN et al.

So lets work through these, first with how the format of the NCAAT hampered it getting over the hump. As at first it was allowed for teams to actually play in both, and many teams did! Then in the mid 50's the NCAA banned schools from doing both, but that still didn't address the final option, that of selecting to play in the NIT as the preferred option. That is until 1970 when then #8 Marquette said thanks but no thanks to the NCAAT and went on to win the NIT. Following that that NCAA said that any school who turned down an NCAA bid both couldn't play in any other post season event, but would be ineligible for a bid the next year. Additionally until the mid 70's the NCAA had rather strict limits on how many teams it would select from each region and would only pick one per conference by and large. That means since the NCAAT at the time only had a 25 team field, the 16 team NIT field still had plenty of VERY good teams to pick from including many in the top 10 in the polls many years. But come 1975 the NCAAT field is enlarged to 32 and the At-Large era begins with multiple teams from the same conference now being invited. This then really was what killed the ability of the NIT to attract teams that would create nationally interesting games.

So we have addressed basically the first two factors, that it took several decades for the NCAA to stop limiting itself, and usage of its power as the body regulating the sport to boss around what in the end was an even managed by a group of schools.

The final one then gets into the realities of pre modern sports media. The fact is few teams had fulltime beat writers, and even fewer got to be on TV regularly. Making what coverage they could get all the more valuable. And that is where the NIT could shine. Before the late 70's all NIT games were played in Madison Square Garden, right in the heart of that Mecca of media that is NYC.

Meanwhile the NCAA played its first two rounds on campus sites still, further highlighting the difference in the experience both for the teams and recruiting benefits. Consider for instance the interesting 1966 Basketball season. It was one of only 2 times the entire NCAAT was played on selected college sites, with the final 4 being played on UMD's campus.

The East Region then got to take a long drive 4 hours from anything into the hills of SWVA to Blacksburg and the campus of Virginia Tech. Where Davidson, Duke, Providence, URI, St. Joe's, and Syracuse would fight it out at Cassell Coliseum. This was made possible because the Hokies were out of town enjoying the Big Apple due to their NIT bid.