I've been considering whether, with the state of my house rules, I even still need multiclassing to get what I want from it. My goal is to conceptually approximate AD&D demi-human multiclassing, because that's an important part of D&D to me.
It's not important to me at all that every potential class combination can exist. It's only important to me that certain ones can. Those basically amount to the core four (and their substitutes) with the others. So that means I don't need any combinations of fighters, paladins, and barbarians with each other. I don't need combinations of wizards, sorcerers, and warlocks with each other. No cleric/druids. Monk, warlocks, and possibly bards don't need to be able to multiclass at all.
So what do I need? Fighter/thief, fighter/cleric, fighter/mage. Thief/mage, thief/cleric. Cleric/mage. Along with the possibility to switch some of those for alternate classes, that's what I'm looking for.
What options do I have for that out of the box?
Anything/rogue is super easy, since all you need is some skills and thieves tools, and probably an Expertise or 2. You can add that to any class with starting choices plus feats. Although off the top of my head, I'm not sure all species can get an Expertise feat without my house rules.
Anything/fighter is almost as easy. Everyone gets the same attack bonus, so that's not an issue. All you need is to pick up some weapons and armor via feats and there you go. The Tough feat is almost guaranteed to get your hit points up to par. There are even feats for a fighting style and Battle Master maneuvers. (And I'm looking at 2014)
Anything / wizard or cleric aren't necessarily very satisfying. Because of the way spellcasting works, just picking up Magic Initiate and maybe Ritual Caster feels way more like you are just dabbling.
So now we get into specifics.
Rogue/Wizard can be done fairly well with Arcane Trickster. Add feats if you want lean into it harder. Or just wizard with skills/feats. I never really saw a lot of conceptual sense for a more even split here, so that works for me.
Cleric/Wizard has the Arcana domain, and taking Magic Initiate plus Ritual Caster leans into wizard enough for my needs. Going the other way, just taking the Magic Initiate feat with an appropriate story and a clearly clerical spell (like cure wounds) to up cast with your slots feel like it meets the concept well enough. Arcana domain makes more sense for priest of a magic deity than multiclassing anyway.
For Cleric/Rogue, Trickery domain plus skills and tools hits the goal about as well as such a fringe concept needs. Going the other way makes even less sense.
Cleric/Fighter has War Domain as a great built-in option, and other domains can situationally be as good or better. The traditional dwarven fighter/cleric gets hammers and axes from his race, so all he needs is heavy armor (yay Forge for Moradin!) Giving a cleric enough martial power to feel like a (traditional) fighter also is well supported with 5e options. Going the other direction wouldn't give you much, and would really just feel like adding a bit of flavor to your backstory or progression.
Now we get to the tough one. Fighter/Wizard. I say this, because it is the one I care the most about. Eldritch Knight is good to feel like you are dabbling in wizard. Bladesinger is a valiant attempt, and works well as the specific tradition, but doesn't fully hit the fighter/wizard itch for me because of the incompatibility of the features with extended Strength-based gear options.
This is where after a couple of years of trying to figure out options that would work, I just ended up making a fighter-mage base class. Full caster, with better gear and hp than Bladesinger, but less wizard stuff, and (2014) Improved Battle Magic as its big feature.
And now...I've just started thinking, "Do I even need multiclassing?" I hate level dips, I don't care for things like fighter/barbarian or fighter/paladin, and combining Charisma casters irritates me. What else do I want that isn't already supported?
Ranger/Druid is something I've always liked the idea of, that doesn't work very well. Even having a ranger go all in with Magic Initiate, Ritual Caster and and that druidic cantrip fighting style doesn't feel like it's doing much. And going Druid with some skills and weapon proficiencies only feels like a druid/fighter, though the TCoE option to burn a wild shape to summon a familiar hits a little bit of the animal companion angle.
So, to get back to what the OP is proposing, I kind of like the idea of adding some feats to enhance select multiclassing options. For what I need, I don't need feats that are beyond normal power, and I don't want a chain. Just ones that could add essential ability combinations not otherwise attainable. So for druid/ranger (and druid is harder to do, so that's the right direction) I'd probably need a feat that adds some iconic ranger abilities. So probably something that gives a weakened version of a favored enemy (and maybe terrain). That wouldn't really be terribly satisfying though. Maybe a better bet would be something that lets you snag a weakened version of Hunter's level 3 feature. Druid with a sword, a fighting style, a hawk companion when needed, and a neat combat option from Hunter. I'm not sure that's entirely satisfying, but I'm not sure if even making a base class would be a satisfying way to get the feel in 5e. Maybe the whole ranger/druid idea just isn't a thing that can really translate out of AD&D (where it wasn't actually legal anyway).
The other concept I've always loved was Ranger/Bard, which has the same problem. It would need a satisfying feat to feel like it's adding Ranger stuff, and that's elusive. Ranger/Bard would probably be the prime candidate for an actual 5e multiclassing even split (conceptually), due to bards not traditionally being full casters
Just some thoughts on multiclassing and how feats can tie into it.