So it’s not a big deal to me.
Personally, though, I don’t like to say “The Y”. Just doesn’t feel right to me. But it’s not cringey, per se.
“The U” does feel more organic because it’s the first syllable in both the words University and Utah. I grew up in Salt Lake and it was very common to call it that. I get that people here think they copied it from the University of Miami, but it was called “the U” back in the late ‘80s, at least, and I’m sure a lot earlier than that.