the OLD SSPX, Classic SSPX was ALSO negative on homeschooling, especially where and when they had set up a school.
When I was in the indult (FSSP and ICRSS), I noticed they were very pro-homeschooling (probably ∵ they didn't have their own schools like the SSPX does). One FSSP priest I knew even regularly attended annual homeschooling conferences.
The FSSP parish here hosts a very large homeschooling co-op (~50 students!). Indult, diocesan, SSPX, and other trad-minded all attend.
What's the Resistance's stand on homeschooling?