From OALD, entrench is defined as:
to establish something very firmly so that it is very difficult to change
Does establish here means to cause to be accepted or recognized?
Edit: I came across the word from this passage:
Some people feel that this term is a bit negative, given the association with hunting and death, and they prefer more positive twists on the saying. Several organizations have even sponsored contests to come up with a new and more animal-friendly version of “to kill two birds with one stone.” However, the idea has become so entrenched in many societies that it is unlikely to fade from usage anytime soon, negative or not.