Future POTUS, unless I misunderstood, the implication of your post is that the difficulty some schools have in meeting payroll requirements is a result of the growing number of families where the father is “in learning.” The fact is that schools have had this problem since time immemorial, and in fact the problem is getting smaller, both in terms of Rebbeim’s salary, and in terms of Rebbeim actually getting paid. I am happy to note that the (out of town) school in which I am fortunate enough to be a Rebbe has not been late with a single paycheck in at least five years (although the salary is not quite up to par with in town schools).
Gadolhadorah, Rav Moshe Feinstein was asked this exact question. His ruling was that, in fact, a Rebbe is not precluded from taking a job somewhere else. However, as long as he is not employed elsewhere, he is required to teach even when not getting paid. This is because, since it is prohibited to get paid for teaching Torah, a Rebbe’s salary is classified as “Schar Batalah,” meaning he is getting paid not to take another job. Once otherwise unoccupied, the responsibility to teach devolves naturally upon him. Therefore, a Rebbe is free to take a job elsewhere, but he is not allowed to strike.