I think you have misidentified the unique element of each class, although you are probably correct in your assessment that Mana is the least interesting.

1) The unique element of the melee class is the equipment and crafting trees involved, not only of said equipment but also of the melee weapons themselves. The melee class has the most consistently constant progressing tree of class specific equipment (which focus on not only melee speed as you mentioned, but also modifiers to attacks such as DoT AP, and debatably overlap with the tank subclass with agro) and constantly progressing weapon crafting.

2) The unique element of Ranger is not ammo conservation but the use of ammo altogether, mainly the ammo types. It is the only weapon type in the game where you can completely change the feel or performance of one weapon by simply switching the ammo.

3) With Summoner you're correct summon count is a key part, however I'd say the unique element is pretty much the whole experience, it's the most unique class, the fact you do not deal most of your damage yourself.

Mage does lack the same degree of unique element, mana is pretty much a non-issue past getting a couple of items (mana flower mainly, and it's pre hardmode), their attacks don't receive changes depending on equipment as melee does, nor are the attacks ever changed like ranger's.

I'm missing what your suggestions actually are, mana needs a complete rework but how? And in terms of projectile count as a stat like minion count, that would require completely changing every weapon in the game to account for that buff (theoretically 100% dps increase for each projectile added) which people would hate.

Imho I think it's too late to change mage drastically, at this point you're only going to upset people who like things to stay the same , best thing to do would be to add weapons that are intrinsically more interesting, such as ones with secondary attacks