Thank you Rajat, im sorry for the inconvenience!
I have the same problem on every form of this website (Eg. newsletter subscription in the footer or contact form):
https://www.lavishapt.com/richiesta-informazioni/
While if you compile the form it says that the entry is correctly sent, it doesn’t register the form entry on Flamingo, nor it sends the corresponding emails correctly (they don’t even appear on an email log, so it’s not that they’re not received. It’s that they’re not sent). I also tested configuring the SMTP for the email address that sends the email but it didn’t fix the problem. Test emails (not by CF7) are however correctly sent by WordPress.
I also tried all the troubleshooting in the OP without success.
Seems like something is broken before the email is sent.
Any ideas?