Ask HN: Do you give your email password to your hosting provider?

Network Solutions previously hosted our work email, but several month ago we moved to a different mail provider. NS remains the registrar and web host. Today, several emails went missing and we figured out that they were ending up on the NS mail server somehow. None of the MX records point there anymore, so I called their customer service to see what might be going on.

Both the 1st level rep and a manager requested my email username and password, which I refused to give. On the one hand, I should be able to trust my registrar, b/c they could easily wreak havoc even without my password. But on the other hand, my username and password were certainly not required to solve the problem (maybe look at the headers of the mis-directed emails first?)

The experience left me severely unimpressed with Network Solutions. but perhaps I'm over-reacting? Would you give your email username and password to your hosting company?

(solved the original problem without NS: all of the missing emails were traced back to one user, whose email client was still using the NS smtp server, which apparently doesn't pay attention to the current MX records)