I like to imagine the BDSWAW half of this argument as a kung fu movie and the magic user half as an action movie.
In a kung fu movie each punch and kick has meaning. This is because each motion is indicative of a particular style. For example if you see Jackie Chan swaying drunkenly you know you're in for a drunken boxing movie; it would be weird if he whipped out a sword and started cutting fools down because that isn't indicative of drunken boxing. However in a movie like Crouching Tiger Hidden Dragon, this is totally cool because sword fighting is indicative of that particular branch of Chinese martial arts.
5e handles its multitude of magic users the same way. You won't expect a bard to do non-bard things (or is it un-bard?). Each class has it's particular brand of magic and they stick to it like a martial artist sticks to his preferred style.
The magic user of lotfp or b/x style games is more like an action star. Action movies pay no heed to styles or martial arts. The motions are divided into simple chunks. Punch, kick, shoot, stab, etc. No one watching an action movie will care if Arny punches a dude with a first shaped like a crane; it's more about the punch than the fist. Likewise the magic user has a variety of spells available; this would be his punches, kicks, shoots, stabs, etc.
Where to draw the line? Personally I prefer bards, druids, and wizard/magicians. Bards at least inspire role-playing (I'm one of those dms that makes you utter couplets!) and druids because they can turn into badass animals and have their own distinct flavor far from regular wizards. You could argue the same for warlock and sorcerer but I find their distinction minute at best. (Why? I'm not sure but the authoritarian in me says it's right.)
For my butchering of the 5e spell lists I'll have to collapse the warlock and sorcerer spell lists into the other three. This could lead to some interesting lists. Eldritch bards? So metal