Background
There are times where I feel like a certain spell would very much suit a certain character or concept, but because it's not on that classes' spell list, they can't have it.
The classic example in my mind is that Tempest Domain Clerics can't have Storm of Vengeance (even though it used to be on the Cleric spell list in previous editions) or Whirlwind or other spells that suit their theme. As it stands, you'd need to take 17 levels in Druid (or 18 levels in Bard for their last use of Magical Secrets) to get those spells, at which point you're not really a Tempest Cleric anymore, you're a Druid or Bard with a dip in Cleric.
I wanted to come up with a way to allow certain classes access to certain spells that they otherwise can't get. I thought a feat might be an acceptable cost for this benefit (aside from a DM just saying "just have the spell anyway" or "I say that this spell is on the Cleric spell list for my games", etc).
Proposal
My proposed feat would be that, if you have the Spellcasting class feature (i.e. you are of a class that can already cast spells) or Pact Magic (because Warlocks spellcasting class feature isn't called Spellcasting), then you can gain access to one other classes' spell list and "they are X spells for you" where X is your spellcasting class (if you're a multiclassed spellcaster like a Bard/Sorcerer, then pick one of your spellcasting classes).
So for a Cleric gaining access to, say, Bard spells, the Cleric would use Wisdom rather than Charisma; this idea is based somewhat on the Divine Soul sorcerer, who has access to all Cleric spells but uses Charisma to cast them.
Question
My question is: Is this balanced? If this is overpowered, would restricting it to only a school or two from the other spell list help to balance it? If this is underpowered, I'd probably throw in a +1 to your spellcasting ability to help balance it. Or is this so broken in a way that I haven't anticipated that I just shouldn't do this at all? Perhaps this is something that is only unbalanced if Warlocks are included (so simply excluding them from this removes a bunch of balance concerns)?
Also note that, although I've recently asked questions about Rangers using the Wizard's spell list, this is actually nothing to do with that. However, I am currently assuming that Paladins, Rangers, Eldritch Knights and Arcane Tricksters are to be included in my proposed feat. Any or all of these can be excluded as I suggested for the Warlock above if this is unbalanced.
Best Answer
It's Overpowered
It's not necessarily quite as bad as it looks at first, but there are still significant issues.
Class-dependent factors: this is way more powerful for Cleric/Druid than it is for anyone else, as adding spells to the class list immediately makes them available for casting in the moment. It's significantly more powerful for wizards than it is for sorcerers/bards/warlocks because wizards can accumulate all of the spells in their spellbooks over time, and have access to any of them within 24 hours. In one way, it's weaker for warlocks than any of the others because warlocks are pretty short on "spells known" in general, and only have one spell known per level for 6-9. In another way it's more abusable (as called out by the DMG) because warlocks recharge on a short rest, and that can lead to cast/rest/cast shenanigans if you're not careful about which spells to allow.
Comparison with Magic Initiate: Not meaningful. Magic Initiate gives you two cantrips and a 1/day casting of a specific 1st-level spell. Its benefits are only barely associated with the benefits of the proposed feat.
Comparison with Ritual Caster: Depends. For anyone with an existing Ritual Casting class feature, this feat is strictly and significantly better than taking Ritual Caster to open up ritual spells of a different class type, but those people might not be particularly interested in Ritual Caster in the first place. For those without such a feature, it's unclear.
Comparison with Class Features: Here's the real issue. Bards and Warlocks have "these spells are now a class spell for you" features that are far more constrained than the proposed feat, and that are still pretty highly valued. In fact, "magical secrets" is very nearly all the bard gets at level 18, and it basically consists of getting access to just two spells from outside their class (plus the two "spells known" slots to put them in). It's relatively weak as a class feature - "magical secrets" shows up at other levels as well, alongside other class features, but the difference in scale between "two spells of choice" and "another class's entire spell list" is immense. Similarly, the warlock has a number of invocations of the form "you can use pact magic to cast this spell once per long rest". They're not particularly good as invocations go, but invocations are still relatively rare and precious things.
Fluff issues: When you hand one class the spell list of another class, you destroy an enormous amount of class differentiation. A druid who has suddenly added the wizard spell list is radically different in implications from one who has not. Their powers (supposedly granted by nature) now include magic missiles, rays of disintegration, and creation of the undead. If they then add in the cleric spell list with an additional feat, the implications become that much more bizarre and extreme, as the powers of nature are now being used to do things like conjure friendly celestials.
Issues particular to the Warlock: the DMG specifically calls out a warning on messing with the Warlock Spell List. The fact that they can recharge in short rest rather than long rest makes certain spells suddenly far more abusable. This messes with the Warlock spell list to a degree that it's guaranteed to enable every single one of those abuses. That's a bit more of an edge thing. It could be solved by, as you suggested, not including Warlock in the mix. It's far from the only issue, though.
Final answer: there's no one thing that can be pointed to and say "no, absolutely, this will break everything forever", but there's a lot of signposts along the way that strongly imply that this is significantly more powerful than intended for a feat, especially for clerics, druids, and wizards, it has some potential for actual game-breaking in the hands of warlocks, and it has some significant fluff issues.
Suggestion: instead of making a generic feat for your issue, make a far more focused specific feat for it. Pick 5-8 spells out of various spell lists that fit the theme (including one or two good clerical storm spells in that 5-8). Call it something like "child of the Storm". Then any caster who takes that feat gets those spells added. Having it be limited in scope like that will make it a lot less of an issue thematically, and will bring it more in line with existing options as far as power as well.