If I collect money from people and send it using a licensed money transmitter’s app, is that legal in the US—and can I charge a fee for doing it?

Yes, this is clearly prohibited and illegal on two fronts.

First, when you collect money that doesn’t belong to you, i.e. you are neither the originator or the funds, not the beneficiary of the funds, but are acting as an intermediary, you are a money services business. Because you can have access to the funds that don’t belong to you, you will be classified as a money services business and hence require a money transmitter license.

Here is a video I made, that explains it:

https://www.youtube.com/watch?v=...

On the second front, most apps like PayPal, Venmo, Square Cash, etc. prohibit you to act as an intermediary when you are mimicking the services of a money services business. Their Terms of Service clearly mention this under the prohibited services.