And as you point out with the "forgotten password dance", access to your email account gives an attacker access to your hosting account anyway (although not stealthily).
This was the very first contact. Sending me passwords (same for 'temporary ones' after a reset. I consider that a cumbersome process whereas I consider the initial mail just plain "wrong") delivers more than just the text over this channel.
I read between the lines "Here's your password again from 2 minutes ago. File it away" which again is nothing I consider a good idea. The impression is that they write the digital form of a post-it note of the password for my monitor.
Is it a big issue in itself? Nope, probably not. But it certainly ruined the first impression for me and I'll walk away now. I shared that part because others might (or might not) agree and to potentially save like-minded people a registration.