Couple this with the fact that many people use the same password for other accounts, including email, and you have a definite security problem for the users.
I believe the flaw here is that it sends your existing password rather than sending you an email link to create a new one. Not really worth a post (and all the ensuing discussion).
Guys, this stuff was known and very well understood 30 years ago. Please, please, please stop trying to do security architecture if you don't understand it. This is the reason web security is as bad as it is. If you aren't 100% sure you understand the thread model perfectly, just follow the advice you read from trusted sources on the web. I'm not aware of anyone sane that is recommending storing unhashed/recoverable passwords.
Also, it may sound defeatist, but I think that if a determined, highly skilled and competent hacker wants to fuck you up and steal the passwords, they will. Even if you salted and hashed all your passwords, there's always brute force, sniffing, etc. If they're on your servers reading your code, you're pretty screwed already.
2) When you send emails to a user who hasn't logged in for a while, you can include their details so they're that much more likely to actually log in
Not being able to do these things costs users.
Look, just read a book or two on the subject. You don't have to believe me, but please stop arguing about stuff you clearly don't understand.
Brute force is a plausible attack against anything. No, doofus, I don't mean that you sit there and try to decrypt the SHA-1 - that's impossible. Instead, once you have the salt (which you can have easily since it's in the code which we've already surmised you have access to), you start with the dict database, and then various crack databases out there. In comparatively little time, you're likely to have cracked some 50+% of the passwords in that salted, SHA-1'ed database.
Unless, of course, the cretinously short-sighted designers of the system were so up their own arses about security that they put in all sorts of rules about what you can put in your passwords, requiring symbols, numbers, and no recognisable words, etc. In those cases, we can safely assume that the system is indeed secure - it's also secure from quite a sizeable percentage of users, who won't bother themselves with it.
2) Who said anything about HTTPS transactions? 99.9% of the logins are not HTTPS'ed. If I really want to sniff your fucking password that you use for everything, I'll sniff it from one of those other sites that don't have HTTPS login forms.
Now please stop assuming that everyone is as dumb as you and understand that there are people out there who are vastly more skilled at cracking systems than you can contemplate, and they will fuck you sideways should they really want to. And no, there's nothing you can do to protect yourself against them, other than not be online.
This should not be so surprising. It's the same offline. If someone really wants to kill you, particularly if you don't know that they do, and they're skilled enough at it, you're dead.