I also think that the NY Times puzzles are very high quality.
You may be right that higher payments would result in even higher quality puzzles. There might be a lot of talented people out there who could create some great puzzles, but who aren't even interested in trying because the compensation is too low.
At first glance, it does seem like high quality puzzles are connected to compensation. I think the NY Times pays the most, and they have the best puzzles (IMHO). But that might also be due to the fact that they have a very good editor, plus it's prestigious to be published in the Times. So it attracts some very talented people without having to pay very much.