This is a pet peeve! When Jeff@xyzsite.com sends me a welcome mail with my username & password mentioned in it, I have basic doubts about the site's security measures & privacy policy. Especially when (foolish, I know) I've used the same password on half a dozen other sites. Am I being paranoid?
It's funny, because I don't think you're being paranoid - but note that almost every site is in exactly the same boat on this. It doesn't really matter that most of them don't send your username & password to you in an email; most of them do allow you to get your username and reset your password using only email verification. So even if your username/password wasn't sent in an email, someone who has access to your email can get them.
I think that's a risk, but it's hard to see another way to do it; people forget passwords, unfortunately. It's a fact of life.
I think mentioning the username and email ID is a lot more preferable - and safer. I'm not as concerned about my email getting hacked (in which case, I have bigger problems) than my password being up there in plain text. Most of us sign up for more things than we can remember individual passwords for; many of us probably use a set of few passwords across sites - with our own rules for what's used where. The fact that any person can assume a reasonable degree of password reuse and try my password on (say) Gmail, etc is very disturbing.
While not a solution to the insecurity of the site you refer, you really should check out Password Gorilla (http://wiki.github.com/zdia/gorilla/). Using it you don't need to remember different site passwords, and you never have to use the same password twice.
Don't most people? I have 6 passwords I normally use- with one for untrusted sites, one for money related stuff, etc. Even though the damage is reduced if someone lays their hand on my 'untrusted site password', it is nevertheless disconcerting. Time to use Keepass or something similar, I guess.
I think that's a risk, but it's hard to see another way to do it; people forget passwords, unfortunately. It's a fact of life.