The problem with interjecting an implication that many worlds is true (regardless of how persuasive the arguments in its favor are) into a layperson's explanation of why quantum entanglement doesn't permit FTL communication are threefold:

1. It doesn't have anything to do with the topic at hand. It is, at best, a distraction. 2. It is impossible to design an experiment which is capable of distinguishing between a universe where many worlds is the correct interpretation of quantum mechanics or a universe where Copenhagen is the correct interpretation. Arguing about Copenhagen vs many worlds is literally no different than arguing about Jesus vs Mohamed. We simply do not have the tools to arrive at the truth of the matter. 3. You guarantee that everybody is going to stop discussing entanglement+FTL communication and will start arguing about interpretations of quantum mechanics. I'm not sure if anyone's started talking about De Broglie–Bohm theory yet, but it'll... oops I just did.

I say this as someone for whom many worlds feels truthier.