Okay young one, we have a few things to get through.
First, you should feel whatever you do feel. It’s not as embarrassing to have your parents help you than you might think. You’re in your early 20s, which means most of your life experience comes from when you were a child and embarrassing parents were a social death sentence. You’re an adult now and to a lot of older adults stuff like that becomes insignificant compared to other things. That said if you feel embarrassed, you feel embarrassed, there is no should to feelings.
Two, I suggest you tell your Dad that you appreciate what he’s trying to do, and that you’ll go on the date (because I think you should). You need to tell him that doing this without keeping you in the loop made you feel uncomfortable. He’s trying to help and it comes from a good place but the execution was a little off.
If he doesn’t listen or it still feels wrong you have to tell him that too and ask him to stop. Still I suggest you try it out.
Third, go easy on yourself. Finding a partner is not easy, and at your age you’re going to make mistakes, big ones but that’s okay. That’s what living is. Just make sure you don’t an STI or you or someone else pregnant. Unless future you is trying to have a kid, in which case good luck!
No, boomers don’t go to therapy because they believe it’s for weak people. Millennials don’t go to therapy because they can’t afford it. Same result, different approaches.