Question re: TAF and salaries

Here's my question in all of the mess that's been happening wrt our OC "search". I have seen it stated multiple times that TAF no longer has power like they used to. But don't they still pay a significant part of the HC, OC, and DC salaries? Couldn't they say they aren't going to pay "their portion" of a hire they didn't agree with? Seems like the TAF leadership is unwilling to exert their leverage. Or maybe there's a piece to this aspect of the deal that I'm uninformed about.