I wonder if your partner is concerned that your kids are not feeling enough pressure, and if so, is there a way for them to feel some pressure without sending them to school? My HE’d kids actually do a certain number of exams, perform in music concerns, and do other stressful things. Could doing stuff like this (if you are not already) be a compromise position?
More specifically, is he concerned that they don’t have the life skills needed to cope with pressure and/or taking exams? Like I said above, if that’s the case then I think it could possibly be an issue in the future, but there are many, many ways to remedy this without school. And as sustainablemum has pointed out, just going to school is no way to guarantee that they get these skills anyway: a better strategy might be to work specifically on how to manage stress, seeking out activities outside your kids’ comfort zones, etc.