As has already been mentioned, requiring buxiban teachers (and presumably kindergarten teachers too) to be qualified would make it impossible for anyone to hire foreign teachers without seriously upping the tuition fees. So, people get what they pay for.
With regard to teachers in the school system, here’s what I can say about that:
I am employed through a recruiter that has contracts for several (maybe even many – I’ve heard it’s a big company, but don’t know) schools in Taiwan. None of the teachers at my school or the two other schools we have regular dealings with is unqualified. I have also encountered other qualified teachers randomly, but employed by our company and at other schools (including private schools). I’ve never encountered a teacher in the government system who isn’t qualified. Maybe some have fake degrees, but from everyone I’ve spoken with, it’s been pretty obvious they’ve gone through an education course in the West.
For a while, our school was using some subs who were not qualified teachers, but they didn’t do that for long. We now have a surplus of teachers at our school precisely so we can cover each other’s classes and so they don’t have to bring in (unqualified) teachers from outside. They do all this despite the fact that it is often extremely hard to recruit teachers here. We have high turnover still and it takes them ages to get new teachers. Whilst they treat us in typically odd Taiwanese ways in some ways, in others, they’re eternally grateful to those of us who have seen our contracts out and renewed them.
Therein lies the rub. For a teacher in the West, coming to Taiwan would not only be a sideways step in a career (unless they taught their specialiations in an international school), it would also probably interfere with settling down and raising a family. Without exception, every teacher I’ve met has thus been one of the following: retired/close to retirement in the West, or at least has grown up children – this entire group seems to make up 1/2 of the workforce; a hardcore traveller looking for a base to explore Asia; a misfit in the West; or has some particular connection to Taiwan (e.g. the pretty common scenario of meeting a Taiwanese partner in the West and then marrying and moving here).
Then, of course, there’s the international job market. I’ve known two guys who have left Taiwan to the Middle East because they could get more lucrative offers.
Making the buxiban market like this would practically wipe out half the people here. You might get some very young people doing it for a year, maybe two at most. However, it would be the kiss of death on getting most girls (and probably a lot of guys) over the age of 25 for any or all of the reasons given above regarding money, career or family.