Is it possible to get into a nice private school without UC regents invites?

I understand your concern. When my daughter applied 4 years ago she only applied to UC’s. Similar stats to yours. She was waitlisted at soo many UCs that this year our son decided to apply OOS as well. So far he has not one Regents or Chancellor’s invite at all. He’s a 4.4 WGPA, 9 APs, SAT 1400 (that’s his weak spot), plus Eagle Scout and 12 years of robotics with leadership positions. But he was admitted for engineering to Virginia Tech, Purdue, and WPI so far. I think the UC’s have just been flooded with apps and there is just no predicting who’s in and who’s not. So while he is still waiting to hear from UCB, UCSB, UCI, and UCSD at least he knows he has some great options already locked in. (but they are expensive!)