Reading a bit between the lines, particularly some of the released notes and interviews from Herb and others who worked behind the scenes (look up Herb Solow on YouTube, there's quite a few of them). But TOS was sure to be cancelled at the end of season two, and Gene did not want to leave the show - he had nothing else to turn to and hadn't sold another show (and, not coincidentally, wouldn't until he went to syndication for TNG).
I've seen a lot of those Herb Solow interviews, and I still don't know if I buy your point. Roddenberry had a five year development deal with Desilu/Paramount which lasted from 1965 to 1970, so he wasn't facing unemployment if the series was cancelled in 1968. On the subject of cancellation, the series was
never cancelled after the second year (see the excellent article located
here). The rumors of cancellation at the time were simply Roddenberry's PR machine at work.
Also, Roddenberry wasn't involved as a hands-on producer at all in season two, with the exception of "Assignment: Earth." He was personally involved with that one because he wanted the extra money that would have come from being the creator of another television series if they managed to sell the backdoor pilot (they didn't, of course).
Paramount (which had just bought Desilu) did not want to put up with the crap of 'that damn show' and it's behind-the-scenes reputation. The extremely large budget was slashed, a push for marketing was made (hence Chekov's sudden rise to prominance), and a less "troublesome" producer was selected. Simply put, Gene was not welcome in Hollywood anymore - which is one reason he felt so safe lying, defaming, and attacking them (though never by specific name) through the convention circuits through the 1970s and early part of the 1980s.
The budget being slashed had to do with (a) the series being more expensive for Paramount than NBC was paying for it and (b) on top of that, NBC slashing how much it paid per episode because it was earning less per episode due to decreased ratings. As far as Fred Freiberger goes, Roddenberry selected the producer himself. He may have never intended to return as a hands-on producer during the third season. Finally, although I believe you've made the claim about Chekov being elevated in prominence before, I don't agree. Neither does Walter Koenig, for that matter, who notes that after "Spectre of the Gun" was filmed (the first episode made for the third season), he saw his role diminish quite a bit.
Roddenberry certainly did burn his bridges in Hollywood, though. In addition to trashing the Networks and Desilu/Paramount (something he was doing in early convention appearances, and in print by 1968, when
The Making of Star Trek was published), his post-Trek projects were all failures. His script for a
Tarzan feature was deemed inappropriate by the production company behind it, and summarily shelved. His development deal with Desilu/Paramount ultimately went nowhere after
Star Trek. And when he wrote and produced
Pretty Maids All in a Row, it was a disaster (and Herb Solow only gave Roddenberry the project as a favor). Roddenberry's chief source of income in the early seventies was the convention circuit.
It's also why he lost control of the movies, and then TNG fairly early on. (Gene's health was the public story for the latter, but Berman and Braga had taken the reigns mid-way in season one, after the first batch of episodes were so badly recieved).
Gene lost control of the movies because he was a terrible movie producer. He constantly re-wrote things (often for the worse), wasn't available when people needed to consult with him, and let the project's budget run out of control. I'm not sure he lost control of
Star Trek: The Next Generation so much as had to leave it due to his health -- which was much to everyone's relief. His constant re-writing (for the worse) of other people's material led to a mass exodus of writers and production personnel during the first season (including just about everyone who was an alumni of the original series).
And, as another poster has noted, Braga wasn't running the show until well after
Star Trek: The Next Generation had ended.