Their purpose is the same as all those other things you mention. To get a character where the player wants it to go via several different options available.
5E has been set up and built such that there are many ways to skin a cat. Because no one agrees on the one right way to make something. Thus, rather than shutting some people out, they instead give multiple paths to choose from...
...Is "Rune Magic" divine magic? Many would say no. The fluff just seems wrong. Thus, making it a Cleric sub-class might not feel right. Would a "rune mage" write all their runes down in a spellbook? And get access to all the different types of magic open to wizards? Eh, maybe? But maybe not? Thus, making it a Wizard sub-class also might not feel right. Would "rune magic" be a single thing with several small abilities you'd get, the same way you do the Magic Initiate feat? Again, maybe. But does it feel like it gives rune magic short shrift if all you get from it is ostensibly the power equivalent of one 1st level spell and a pair of cantrips? To some, maybe yes. And thus, making "rune magic" just a feat might feel like you're short-changing the concept.
So if it doesn't really fit under the umbrella of a certain class's fluff or mechanics to be a sub-class of that class... and it warrants having more mechanical heft than what you'd get a from a single feat... BUT... the concept *might* be more limited that making a full 20-level class out of it seems potentially like overkill... what is left? To me, the answer is either being making "feat trees" (where you can take several feats in a row that gain more and more power), or you create the idea of "microclasses"... concepts whose fluff and story (and the game mechanics which reflect that fluff and story) are individual enough that they deserve their own "class", but perhaps don't have enough heft to them to make 20 levels of it. But only 10 levels? Only 5 levels? That might be an idea worth thinking about. Which is exactly why I think they broached the subject with us in this UA. To see how we feel about this middle ground between the feat, the sub-class and the full 20-level class.
Let's take another example: the Artificer. Most people I read seemed to agree that having it as a sub-class of the wizard did not feel like it was the best way to do it. Some folks would be okay in the long run with it should WotC have decided that was the way they were doing it... but other folks were giving all manner of reason why it shouldn't. A lot of the wizard's fluff and mechanics didn't really reflect their idea of the artificer. Thus some were making the artificer a Warlock sub-class, a Bard sub-class, or a Cleric sub-class-- but in almost all cases they were using them purely for the mechanical expression of the artificer concept, and were stripping the actual fluff off of the main class. No one really thought an artificer was actually a Warlock-- making a pact with a greater crafting entity. LOL. Some other folks felt it deserved to be a full 20-level class itself, because that way they fluff of the class and the mechanics of the class could be created out of whole cloth so that it would all work together seamlessly. And granted, there's some truth to be gleaned by that.
But is an artificer-- a magic user that puts spells into objects for people to use-- closer to a wizard than we might give it credit for? As an actual sub-class of the wizard it might not work out (because of all the baggage and features you get as a wizard that don't apply to the artificer)... but conceptually is an artificer using the same scientific concepts of magic that the wizard does-- the same method of arcane textual usage-- but is just putting that magic in a different form such that the artificer and the wizard are two sides of the same coin? To the point that you could (if you wanted) make a artificer Prestige Class-- an artificer of say only 5 or 10 levels-- that would make sense to multiclass with the wizard? You start as a standard wizard, going through your first several levels of the Apprentice tier learning the basics of magic-- and then at some point turn away from standard wizardry to begin your work as an artificer or alchemist? Where you begin inserting your magic into objects, rather than casting spells. And you follow that path for maybe 5 or 10 levels with fluff and mechanics that are running parallel to the wizard, but are not OF the wizard? Might not that be a possible middle ground?
Now let's be fair here... I'm just spitballing. Perhaps the artificer/alchemist "magical object maker" concept deserves and can support a full 20 levels. Or perhaps the "rune magic" concept can support 20 levels. Or the "tactical warrior" warlord concept can support 20 levels. And thus the idea of the middle ground "prestige class" isn't necessary.
But maybe it could be? Maybe a warlord prestige class-- 5 or 10 levels of the warlord that is build to be a possible fighter or rogue multiclass might work? Or maybe not. I dunno. But it's certainly a possible option, and a different way to build a character concept beyond what you could build using feats and sub-classes.
It's interesting to think about, no?
