That really doesn't have anything to do with the particular topic of TOS in syndication and is more your own opinion than anything else, but what subsequent Trek shows definitely did was expand the Star Trek Universe. DS9 and VOY were both spinoffs of TNG, with ENT as a prequel to everything.TOS wasn't just syndicated in the US but in sixty other countries (where it was translated into other languages for some). It was really after the show's cancellation that it got its biggest following. It's impossible to say if it had a bigger audience at any one time after leaving NBC because it often aired at many different days and times in syndication, but definitely many more people saw there due to its greater presence in both US and international TV markets and its length of time there.
Yea, but if it's quality is bad you can be sure it'll never be heard of again. It's hard to create a universe and the other shows just rode the coattails of GR's creation. Don't think that the spin off producers weren't thinking they were doing something much better than TOS, and were more clever than he was, etc.. 'We know you mean well, Gene, but..' Many have tried and died.
I didn't realize that the fact that TOS was a high quality show was just my opinion.