Answering the "is the private school worth it?" question requires information that isn't provided. Given what we know, the only reasonable answer is "Maybe!"
All the detail about her present financial circumstances is irrelevant. And you can stop worrying about her life going to hell in a handbasket. She's fine.***
The only question that matters is:
Is there a near enough public school that meets the child's needs?
One can never assume. Often the reason the non-public alternatives still exist and parents are willing to pay for them is that the public/charter options are abysmal. If your local school sucks, it's your kid's life and future that is at stake. It's your kid, and you can't wait years for the schools to turn around once all the financial, political, infrastructural and social investment in improvement finally occurs.
Fifty years ago, my folks sent their first few kids to public high-school. In those days, the place was pretty rough. A few things happened that I won't go into, but the upshot was that the youngest seven of us had no choice. Mom put her foot down. We were going to Catholic high-school. It was an extreme financial sacrifice. Extreme. My folks were frugal and we didn't starve. There weren't the need-based scholarships that exist today for parochial school. You paid or your kid went elsewhere. We weren't a low-income family, but we lived like one so they could pay the tuition.
My husband, who really was poor, had far more material comforts in his childhood because their low-income meant they qualified for free breakfast & lunch, food stamps, heating vouchers, etc. (I could do a decent rant about the Christmas presents & the heaps of toys they got). Of course, he also went to that local public high school and suffered for it. My Dad's income was solidly middle-class, stretched by having lots of kids and then further stretched because they spent so much on tuition. In the going-thru-it years, it did feel like we were deprived and that other kids had more. In hindsight, I'm very grateful. In the case of my local school options, Catholic was superior in several ways. The education was better, the opportunities were better and it was safer. But it's a hyper local question.
I can read about Virginia Walden Ford's efforts and recognize my mother and her mother before her who moved heaven and earth to put my Mom in a Catholic girls' high-school. We're also Catholic, so there was an ideological alignment.
For my siblings, some ultimately chose the same Catholic HS for their kids, some used the (by then much improved) public school or nearby public schools from neighboring towns for their kids and grandkids. And some went very fancy, sending their kiddos to really chic prep schools. Any school can still be a bad fit. Sometimes families need to pay when they find the right one.
Your friend will be fine. Her son may or may not thrive in this Catholic school. Personally, I think it's pointless if they aren't Catholic. But if they are, and they belong to a parish, there is often financial aid to help further a Catholic education if the tuition represents a genuine hardship.
***She's fine.
I say this with full confidence because 100% of the time when someone presents an untenable or unsustainable financial situation, they have a rabbit in their hat. Or they have a trust, or an inheritance or a sugar daddy or something that didn't factor into the first dump of information. It's hinted at here with "father helps with the mortgage" which tells me that the teen's grandpa has enough to help his daughter. Should grandpa subsidize a lifestyle that includes private school? What would he rather do with his money? These are rhetorical. It's not our concern.
She's fine.
You are allowed to care about her, offer advice and make suggestions. But for your own peace of mind, figure out how to not let this bother you. If taking on this expense turns out to be a giant mistake, it's not going to be her last one. It's clearly not her first.