The rules for entry into countries are set by the relevant regulatory authorites, not the airlines. The airlines are just the ones carrying out the document checks, and thus liable to validate them before accepting you on board.

If the rules for entry into France from the US dictate no PCR test is required, then DL will not ask you for them. Just proof of vaccination.
When travelling to the US, an antigenic or PCR test is required. The good news for you is that it is quite easy to get tests in Europe at relatively low cost and no hassle (around 30 € for an antigenic test) which will be fine for travel to the US.

You're responsible for keeping up-to-date on the entry requirements ; don't expect the airlines to send reminders.