If you put the game mechanics aside and just look at them thematically, the similarities between the arcane classes far outweigh any of their differences:
A wizard is a mage who studies different schools of magic, and carries a book of spells and a familiar.
A sorcerer is a mage who studies different schools of has a bloodline mixed with magic, and carries a book of spells magical bloodline and a familiar.
A warlock is a mage who studies different schools of sold their soul to be able to use magic, and carries a book of spells, a talisman, a sword, and or a familiar.
An artificer is a mage who studies different schools of magic technologies, and carries around a book of spells set of tools and a familiar automaton.
So are the game mechanics alone enough to justify having four different classes and about 40 subclasses? I don't think so. I'd prefer to have one unified set of mechanics for all "Mages," and then put all of these different themes as subclasses under it. For my nickel, I'd want arcane spellcasters in my campaign to have a small selection of spells that reset on a short rest, so I'd use the spellcasting framework of the warlock as my "Mage Class," and then put all five dozen arcane spellcasting subclasses under it.
Your description is pretty reductive. (For instance, sorcerers aren't all about bloodlines, specifically, and Warlocks haven't necessarily sold their soul). I for one, like how the playtest document describes the theme of the mage classes. Have you read them? Are you dismissing them wholecloth? (I'm not going to copy/paste them.) But below are my truncated versions of the descriptions.
A Wizard is a mage-scholar that studies and masters the ambient arcane magic of the cosmos. They follow magical traditions to discover, create, and collect spells into their spellbook, a personal tool that houses their collection of magical knowledge, which is also used to cast ritual magic easier. Mighty wizards learn so much about magic, that they can ultimately modify and create their own spells into their spellbooks.
A Sorcerer is a mage with innate magic imprinted into their very being, with some knowing how or why, and others having no idea. Sorcerers don't "learn" magic in a traditional sense, rather they figure out how to harness the raw energy within into the existing patterns of magic in the universe, and as they become attuned to their own power, it grows. They can manipulate that raw magic when it is being cast, to enhance it in a variety of metamagical ways, and in time the origin of their magic may manifest in wondrous ways, altering them forever.
A Warlock is a mage that has no inherent magic themselves, and rather than study it all on their own, they make pacts with otherworldly entities, colloquially called "patrons," to acquire magical power. This power often comes in the form of ancient secrets and invocations that many ancient mages left behind for the independent traditions of Wizardry. Warlocks don't necessarily venerate their patrons, but they do play a dangerous game treating them as a means to the Warlock's own ends, amidst the ineffable machinations of those patrons. There may be a price of service for the warlock's new power, but often the use of the magic itself empowers the otherworldly being or furthers their agenda.
I think these are pretty different thematically, and the new incarnations of the playtest descriptions do a good job differentiating them. Despite my appreciation for the new class-specific spells, I also like the richness of all the non-spell abilities that differentiate them, and distilling them all into one class does not give design room to recreate them, whether thematically or mechanically.
