Earlier quoted context omitted.
Checks? France loves checks and use them more than Americans. No public health care? That isn’t true either. Medicaid and Medicare are public health care. Another legitimate peculiarity of Americans is the vastly higher disposable income compared to most other countries. Even with all of the “free” stuff, the French have lower wages and far less disposable income than Americans.
The French have the RIB system, wherein you can send a snapshot of what your account details are, and the bills are automatically deducted from there. In practice, that means you need to use cheques MUCH less frequently than in the US. What's worse, the US doesn't have an easy way to transfer money electronically between banks! Thus requiring even more cheques.
When living in France I never came across the use case you describe, at least not in this form. But I did often come across people wanting money from me sending their RIB as an image attachment to emails rather than copying and pasting their IBAN into the body of the mail like elsewhere in Europe.