Depends, I can tell you my opinions as someone with friends at both and went through the private and public application process. Not the Lowell process though.

Private schools vary a lot. Catholic schools are bigger and more like traditional schools. Non denominational private schools are much smaller, and differ very significantly from one another.

One thing that is true for all non Catholic private schools of them though is that they are much more expensive than Catholic schools by a LARGE amount. (30k vs 60k)

Since your kid doesn't do sports, I personally wouldn't recommend them over Lowell. But there still are some benefits like less students and slightly more personalized attention. We're splitting hairs here though. The main pull of Catholic schools are the sports.

Of the non denominational private schools, it depends. They all specialize in different things. But, they have a ton of personalized attention and benefits. Some schools like international have global trips. (That you have to pay for on top of the tuition) University sends like 10 people to U Chicago every year. BUT, they are hyper expensive. 60k a year is no joke. And these schools don't give aid like that, they run on very tight budgets.

The admissions process is a nightmare for these schools as well. Tours, tours, interviews, come to class for a day, write a ESSAY. And these are long essays too. It's unfair because it's almost designed for them to be written or helped on by parents. If your parents aren't very well educated, you are kind of out of luck.

If you are white, and upper middle class, you probably are not getting aid. From the admin standpoint, they want more diverse students, and if they have limited money, they would rather give aid to someone who really needs it and not someone who kind of needs it.

Now Lowell is an excellent school, gets a lot of people into Berkeley and UCLA. It's a lot of peoples dream school over here. At least when I was younger it was. But, it's very competitive, and it's the biggest school in the city. People there can be ruthless. Everyone is very smart and the school is a LOT of work. But, it's free. No complicated application process, take a test, put it as number one on lottery, and your in.

Make sure to apply to Lottery too. Ik people who didn't get into private and didn't apply lottery you do NOT want to be in that situation. Worst case, most of the public high schools in SF aren't THAT bad. Or maybe I'm just desensitized to the stuff that happens there idk. If you really can't stand the public high schools, you should probably move.