Any USA catholic who sends their kids to public schools in this day and age, isn’t catholic and certainty isn’t a trad. Assuming we are comparing homeschool vs private sspx/trad school then...
As far as money goes it’s probably cheaper to homeschool so those that don’t, aren’t saving $ by paying for an sspx/private school. But...some private schools are free (or tuition is provided by donors) so that’s a factor.
It also probably takes less time to homeschool than regular school, if you factor into commute times, homework time after school and all the other wasted time that a child spends on activities which a homeschool can’t provide (school plays, field trips, etc).
It’s much more efficient to homeschool, and pick up teaching where you left off the previous day when you can adjust teaching to one student vs 10 or 20.
So, obviously, if parents aren’t saving time or money by sending their children to private schools, why are they doing it? Because either 1) the mother doesn’t have the confidence or patience to teach, 2) the mother isn’t intellectually inclined enough, 3) the mother isn’t organized enough, 4) don’t know enough about homeschooling or how it works. 5) Don’t have a network of support in the homeschooling community.
Homeschooling is like running a business. Gotta be organized, disciplined and able to solve problems on your own. And MOTIVATED EVERY DAY. Not everyone can do it, even if they want to. Not everyone has the personality.
Even 100s years ago, when private schools weren’t as available, homeschooling was rare. Most kids who grew up on farms had minimal schooling. Even if the town had a schoolhouse, kids had a few years to learn the basics, at most. Homeschooling as it is now, especially with the breakdown of the nuclear family and with couples not having the advantage of living close to relatives, is a luxury because it falls on one person, the mother, to make it work.