Clerics are an accidentally-created class that have been a mess in in every edition and after 2E, been a borderline-OP mess. And yeah the line between them and Paladin has always been a confusing one. But I don't think D&D needs to be looking at drastic changes here, because at this point, it's become self-defining.
Sadly, yes. I would probably prefer a "white mage" or a unarmored/lightly armored priest class. There are obviously ways to do that in 5e (e.g., Divine Soul Sorcerer, Celestial Warlock, etc.), but they are not entirely satisfying ones for the archetype. I am glad that 4e began to walk back the heavily armor cleric a little bit.
The time for drastic changes was 3E, essentially, and Monte flubbed it - which is why he tried to re-write history with Arcana Unearthed (which I love, but is basically 100% an apology for 3E, or from another perspective a "What I actually wanted to do with 3E!").
I don't think this is an entirely accurate reading. It's a bit revisionist, and I suspect that it's because Monte Cook was simply one of the most prominent designers who talked about the game. But Monte Cook was not the lead designer of 3e. Jonathan Tweet was the lead designer. I believe there is even an article or forum thread here on ENWorld within the past year or so where Tweet talks about the design decisions behind the 3e Cleric. I don't think that's something that can just be pinned on Cook as if he were the scapegoat of 3e's questionable design choices.
I don't think videogames are a good model to pursue with tabletop RPGs though. Videogame class divisions are serving a different purpose, a lot of the time. Thinking about classes as roles can have some value, but in the end, historically videogames have tended to divide stuff up in order to do things like force people to reroll their character a lot and spend more time playing their game, or to really simplify concepts because they were too hard for players, and obviously I think we all know any non-combat aspects of classes in videogames tend to get stripped away (slowly but surely) and replaced with more combat-oriented functionality, which again tends to point them towards narrow focuses.
Even if video game RPGs are designed mostly about combat, I think that video game designers understand that people are drawn to play certain class archetypes/playstyles and design their classes accordingly. I have been playing TTRPGs for 20 years, which is admittedly not long in the grand scheme of things, and in that time nothing has changed. There is nothing new under the sun. I have time and time again seen players - even those who have years of experience playing pen 'n' paper TTRPGs - say they "want to play a class like X___" where X is a class or archetype from a video game: e.g., "How can I play a Diablo 2/3 style Necromancer in D&D?" or "How can I play a super heavily armored warrior?" Class playstyle and fantasy is important to a lot of players.
WoW, for example, has like 40-ish subclasses, most of which play like entirely separate classes - I think it's quite a good example of how videogames tend to keep separating stuff out and separating stuff out.
The WoW specs are more akin to subclasses. You still get a lot of core class abilities and mechanics. FWIW, I don't think it's that far removed, for example, of the Eldritch Knight and Champion Fighter subclasses playing quite differently or likewise the Swashbuckler and the Arcane Trickster Rogues playing quite differently. Not every class has good subclass distinctions in D&D 5e. (I'm looking at you Wizards).
However, one of the reasons why Warcraft has the design it currently does with its specializations was because people wanted their specializations to be equally viable and the team wanted each specialization to embrace different class fantasies and layered mechanics. IMHO, even if the different specializations of druid, for example, play differently, they still very much feel like playing the Druid, though in the early days of WoW (Classic & BC) there was more form shifting than present.
Talking of Clerics, WoW does a kind of interesting thing, I think more naively than consciously, which is that the Paladin in WoW is basically both the D&D Cleric and the D&D Paladin, pretty clearly, but the Priest is a separate class entirely with is more like a combination of 4E's Invoker, and 3E/4E's Psion, with a bit of a Far Realm theme to the some of the Psion stuff.
Again, see the Starfinder Mystic, which is part priest, shaman, druid, and psion.
If there was a start-from-scratch approach that didn't need to follow the game's legacy, I would consider adopting something like this that distinguished between the heavily-armored Paladin/Arcana Evolved Champion and the lightly-armored, wisdom-based Mystic that you could layer a Priest, Shaman, Healer, Psion, etc. on top of.