Can you use an open source weather app, or is the problem deeper than that?

Yeah, that's true. Often people will take the money from users AND from selling data. But you can (usually) verify if that's the case by checking the app's permissions required.

The point stands, however: you either pay for the software with money, or with data - with the only exception being the unusually rare FOSS project here and there, which either lives in relative obscurity or grows to become large enough for the creators to either start requiring money, or just fold under the load...