About 15 years ago I was teaching at a university in southern Taiwan, with an MA. I applied directly to roughly a dozen universities in the Taipei area, and all rejected me, all saying they only hired PhDs. A year later I applied to the same schools, plus a few more, and one accepted me. So that was over a decade ago - I assume getting a job is tougher now.
I applied whether or not they had openings listed. If I recall correctly, the places that hired me didn’t even have posted openings.
Note that the part-time uni jobs are not good. You’re paid by the hour, roughly $600 - yet you’ve got to plan a course and do marking as well. Most cram school jobs pay roughly the same per hour in the classroom, and involve a lot less outside-the-class work.