I'll say it outright. I would have preferred it if Cat confirmed Kara's secret. It would make their relationship much more interesting.
So she fires Kara. Good. She treats her like shit all day. I don't care how many friends she has in that building, being treated like a body slave by your immediate superior is mentally and even physically draining, even for someone who can bend steel in her bare hands, and eventually you need to get away from that. And it's not like she'd be out of work for long. J'onn Henshaw indicated he'd take her on. That would likely mean better pay plus benefits and hazard pay, and she'd have people much better at keeping secrets helping her protect hers.
As for Cat, she went from somebody who would potentially be intriguing to somebody who's just annoying again.
While I agree it would have been nice to see Cat in on the secret, the character did raise a (surprisingly) good point when she pointed out Supergirl would be better off saving people 24/7 than toiling away as her lowly assistant for much of the day (although if she had any idea how many times Kara already sneaks out every day, I imagine it might not bother her as much).
But whether she knows or not, I'm one of those who still finds the Cat and Kara relationship to be the best and most interesting one on the show. There's always a certain tension there, with Cat being at once a mentor figure to Kara and an adversary, that I just find incredibly fun to watch.
And off course this being a Berlanti show, you know she's going to find out the truth eventually (or find it out again I suppose), so I don't see much harm in postponing that moment til later in the show.