She seems very concerned about the fact she can't remember drinking more than a couple sips of wine and then making out with a bunch of guys.

Kate was concerned about the video most of all. And the first thing an alcoholic will do is lie to hide their addiction. It's one of the primary components of the disease.

I do think it's pretty far fetched to insinuate that Kate is an alcoholic.

I've had the misfortune of living with an alcoholic myself, and I can say that hidden bottles and beer cans are a telltale sign of alcoholism. Without disrespect, the far fetched explanation is that she would be "keeping it for religious rituals" or that it's not alcoholic. The first explanation got debunked by another reply who explained the role of wine in the Church, and for the second one: why would she hide the wine if it wasn't alcoholic?

Kate drinking and taking part in a Vortex Club party (which is otherwise out of character for her as well) are perfectly coherent actions if you assume that she's rebelling against her overly strict family. But the drinking clouded her judgment, and she clearly didn't expect Victoria to record the video and post it online. That's when she started despairing and everything started to fall apart even more. I think all of this follows pretty logically from each other.