However, this doesn't work, and instead sends a blank email.
Actually, it's surprising you get any mail at all.
For a start, you're not piping the output of
grep into mail, you're redirecting the output to a file called 'mail'. I'd expect this to throw an error, so maybe it's a transcription issue and you didn't mean to '... > mail', but you meant '... | mail...'
Once that issue is resolved, you'll find you still get blank emails. Try the command without the pipe to
mail and you'll see why - there IS no admin account in /etc/passwd. Therefore the grep doesn't output any result, so there's nothing to mail.
User accounts are not stored in /etc/passwd for this very reason - it's too easy for someone to grab the passwd file and find the encrypted passwords. From there it's a short jump to cracking the password and compromising the machine. If that's your goal here, move on.