From a game design standpoint, "witch" occupies a much clearer conceptual space than "sorcerer" or "warlock" does and is more immediately familiar and understandable to a new player and the general public than almost any other class. (The general public will give you a blank stare when you mention "cleric" or "druid," for example.)
It might be that witches should have been introduced decades ago, and they've been elbowed out of the picture by this point, but they're arguably one of the biggest concepts in fantasy pop culture that doesn't have a clear specific analogue in D&D. (And no, "you can play it if you take something named something else and then adjust it in several ways a newbie wouldn't know how to do" isn't the same thing.)
For myself, I would prefer it be either a core class (replacing the sorcerer, perhaps, which has really lost a lot of its reason for being in 5E, with the wizard and warlock dividing up its 3E role) or as a PHB-level subclass of either the wizard, warlock or druid.
My wife is playing a transmuter hexblood, which is as close as she can get as a relative D&D newbie to capturing a classic Wizard of Oz wicked/Halloween witch vibe. It's not tremendously satisfying, honestly, so I might see if she's interested in playing a Witherbloom wizard instead. (She's got a real bias against druids after playing EverQuest.)
My issue with this idea is that which witch is which?
That is to say, what measure is a Witch?
Witches are like Trolls: while in continuity with one another, most depictions are in complete discord from each other (All Trolls are Different, so are All Witches). You can list a line of iconic witch tropes and I could list dozens of fictional or folkloric witches that break several of the tropes you have, or even lean into antithetical tropes.
Witches differ from each other most specifically in HOW they came into their power. D&D classes often share overlaps in conceptual space of what their power is and what it does, but rarely overlap in where it came from - how did I get from background to Level 1 class member? If the members of a class got their core powers in completely different ways, are they really the same class?
This is like the classic Magic Warrior trope - most people agree that it needs to be in the game, but nobody can agree what it looks like. A lot of people wanted a half-caster arcane magic user with access to a subset of fighting styles to parallel the Ranger and Paladin, but beyond that nobody could agree what it should look like because the origins of power for Eldritch Knights, Hexblades, Blade Bards, Bladesingers, Battle Smiths, and Wild Magic Barbarians are very much quite different. These belong as subclasses of various classes, showing what those classes can do with a little bit of arcana or a little bit of martial training gives them, rather than to a skeleton class with no unifying identity.
This is the same problem that killed the Mystic class - it was a skeleton on which various Psionic tropes were hung as subclasses that owned 90% of the meat and potatoes.
Witches are similar in nature. You've got studious wizardlike Witches like Hermione who studied really hard but also had to have that magic spark to begin with. Then you've got the historical Christian interpretation of witches who made deals with the devil. Then you've got fairy tale witches who are represented by Hags and other Fey creatures in D&D and might be better represented as a lineage than a class. You've got characters like Sabrina the Teenage Witch who is only half-witch and thus half a member of a witch species - suggesting an inheritance factor for some stories. You've got Wiccan adherents that treat witchcraft as a nature religion. You've got Mahou Shoujo who are just as likely to learn their magic from scratch (anyone can access it with the right teacher) as they are to inherit their magic. And you've got some witches like Kiki of Ghibli fame who need to maintain a belief in themselves and navigate her developing emotions and hormonal changes to maintain her powers as they slip away from her (initially she believes it's because she got distracted from doing witchy stuff).
As you might notice above, any and ALL of the spellcasting classes are reasonable choices for a witch character. That's usually a sign that a new class is not needed but rather that this is a very broad concept that is better applied as individual subclasses of various classes. See Ninjas - we have both Way of Shadow Monk and Assassin Rogue.
Alternatively, they could develop Witch subclasses that work for Bards, Druids, Sorcerers, Warlocks, and Wizards, much like the Strixhaven subclasses do. This may actually be the best solution. Do we think that the Strixhaven subclasses already serve this purpose pretty well?
Bard? Yes, (some) witches should definitely get healing magic, (Witherbloom allows that) but whether they are arcane or divine or whether that's even a distinction that exist is a matter of setting lore.
I don't think is quite true. You grab agonising blast and hex and you're good to go. Sure, you could improve it further, but you don't really need to, with these you will be doing pretty horrid damage.
I really wish there was an easy way to support more varied builds. One thing I did was to let agonising blast (renamed 'agony') to apply to all cantrips, which makes it slightly less idiotic to use cantrip other than eldritch blast, but that still is the best cantrip and comboes amazingly with hex. One thing I considered was changing eldritch blast so that multiple beams aimed at one target become one beam, but that seemed like a too drastic nerf. (I.e. instead of 3 x d10+Cha+hex you would have 1 x 3d10+Cha+hex.)
Pact of the Blade Hexblade Warlock is my go-to 5e Gishy.
Green Flame Blade at low end to spread some damage to secondary targets, pile Hexblade's Curse and Hex on top of a target, and have Eldritch Blast available if I need ranged damage. Generally talk to my DM about swapping out the Accursed Specter in favor of War Magic from the Eldritch Knight, or using an Invocation Slot for something similar in place of Thirsting Blade 'cause Cantrip + Bonus Action attack allows me to do a variety of melee cantrips and still feel like a "Fighter" with that second attack. And it's self-balancing against my bonus action abilities like Hex transferral.
Squeeze Agonizing Blast in there, somewhere.
Before 6th level I'm swinging 2d6+Cha damage with a greatsword and dealing Cha mod to a nearby enemy or blasting one target at range for 1d10+Cha.
After that point it's 2d6+1d8+Cha in melee if they won't let me war-magic it. Or 2d6+Cha twice with Thirsting Blade. And if they do let me war-magic, 2d6+1d8+Cha and 2d6+Cha. Or 2d10+Cha at range.
Oh. And Hex is up, of course. So add a d6 to each of those hits. And probably Hexblade's Curse for another Proficiency Modifier in damage and wider crit range.
Eldritch Blast does pull back into the lead for maximum damage at level 11, but the average damage of the 2d6+2d8+Cha is going to be higher. And it won't pull into the lead if War Magic was made available through some method.
Also worth noting... I don't Multiclass. Pretty much ever. It's an optional system rather than core rules and, honestly, the way the game is built I sincerely feel it's better off without minmaxing class-mixing.
Of course... that's if I -want- to play up the violence.
That character also takes the Criminal Background and I aim to play skill-user-ish. Add in Pact Magic for some decent spells for AoE situations or social manipulation and you get a flexible character that burns at range and tears up in melee but is a bit of a glass cannon in medium armor.