Caregiver seems like the only route, but is not either easy. Asides from proven experience, there is a requirement for proven education, and language proficiency.
The process takes so damn long that in the same amount of time that it would take to qualify for a visa, they could improve their skills, and economic situation in the DR, and maybe then qualify.
But really there is no other way, other than going some convoluted route of trying to qualify for some other category that they don't already qualify.
College courses and second language training will both improve outcomes in the DR, and chances of leaving the DR.
You can check everywhere you want, but these are the facts. It sucks, but that's just how it is.
I don't know who is propagating the myth that "canada necesita trabajadores" but its not entirely true, we want skilled labour, our unskilled labour comes from Mexico and Jamaica, possibly guatemala, DR isn't on that list.