It existed in its earliest form in Old English. Every Old English letter had a name. If it didn't have a separate name it's not a separate letter. It's almost directly analogous to the Greek sigma, which in all positions other than word final looks like this: σ but in word-final positions looks like this: ς

Is the sigma at the end of words a different letter? No. In fact, in many scripts there would be no variant and it looked like this in all positions in lower case: c (exactly like our c or the Russian s). The same goes for the Old English long s, which is not consistently used in all texts or used with uniform placement in all texts.

We don't consider variant forms of sigma a separate letter and so we shouldn't consider the variant s a separate letter. For the record, I the eszett isn't considered a separate letter, just a writing convention. Like the long s (and variant sigma) it doesn't have a majuscule form. All proper letters have majuscule forms.

That's a lot of arguments against it being a separate letter.