There is a related debate in Japanese as to whether emails are events, which means you count them one way, or communications, in which case you count them the other way. They haven't been letters for most people in a decade, which is good, because that means you count them a third way. My ex-boss, however, maintains that they're "fundamentally equivalent to paper", so you count them a forth way.
A single item of mail is a letter (or a postcard, etc.), which can be pluralized. A single item of email, on the other hand, is an email, which can also be pluralized.
In any event, it's time to move from e-mails to emails.
Knuth agrees with you (see the bottom of the page):
emails works because we use "email" to refer to a single email message. Contrast to use of mail, which never refers to one piece of mail.
At one point someone at work who should have known better wrote "eMail packets" so that's what we use. "you have an incoming eMail packet", "check your eMail packets", etc.