When I was in elementary school we were taught Every Good Boy Deserves Fun and FACE. And nobody ever mentioned that it was the alphabet. The first time I put the spaces and lines together, and realized that it was just A-B-C-D-E-F-G, I was totally flabbergasted
Well no, when you put them together you get E F G A B C D E F.
Which isn't the alphabet.
Plus the thing you're remembering is where those letters are on the stave.
Your goal is probably to know every note by itself so you see C and it's C, it's not "err, F.. A...C" [although I read notes relative to others more than individually]
But if you're going to "count" to see what a note is, you need some place to start from that you know and it makes more sense to count the lines or the spaces rather than every one. In maths most of us learn to count in more than ones and twos, even though when you put them together you just get 1,2,3,4,5,6,7,8,9,10.
Besides, why did they make you remember the alphabet? I can see many ways of grouping / ordering the letters of the alphabet that would serve some purpose [splitting consonants and vowels, the order on a keyboard and whatever else] but the alphabet itself seems arbitrary and as equally pointless to develop rhymes and songs to remember it in a specific order in much the same way that you're suggesting the notes on the stave are pointless learning because they are this "the alphabet"
Perhaps that's the point, once you've remembered something entirely pointless well enough, you no longer need the rhymes etc. You reach the point where you ignore the pointlessness of it and see it as a significant ordering.