You are correct. It used to be that R.O.C. immigration would stamp as many passports as necessary when LEAVING the country.
Those laws changed at the end of 2000 (I believe, around that time, but don’t hold me to it) and now you may only have ONE passport stamped upon exit as you stated, though I’m curious to know why this would pose a problem for your two children.
I have dual citizenship and enter and leave the R.O.C. with my R.O.C. passport. Though my U.S. passport shows that I’ve been NOWHERE and looks rather ridiculous to U.S. immigration when I re-enter the U.S. – U.S. immigration doesn’t really question my whereabouts and in particular how I EXITED a foreign country, because if I was able to enter it (in this case R.O.C.) with or without a visa, the entity that would care how I got in and back out would be the R.O.C., not the U.S. – though please note that this is my own rationale and not the official word.
Given that your children most likely do not look suspect and that the R.O.C. is not one of the 7 (? I forget how many) nations considered “hostile” (at least from the U.S. perspective), having no EXIT stamp from a foreign country when re-entering your home country shouldn’t be a problem?