First Answer -- If not Wild-Clasped/Wilding, no

So, there is another layer to this question, which is this:

Do my items work while I'm Wild Shaped?

The simple answer is, generally, no. Unless you get into things like Fangshields druids, Planar Shepherd, Master of Many Forms, or simply ape-ish forms (aka, any form that could feasibly wear the gear in the first place), most of them will not work while Wild-Shaped normally.

To make them work while Wild Shaped, you have to add either

1) Wilding Clasps from MIC (slotless, added to an item/slot, costs 4000 GP)

or

2) Wilding enchantment (only applicable to armor/shields/weapons, as far as I know).

A few druid-themed items will specifically state they work while Wild-Shaped (essentially having the Wilding quality), but Belt of Giant's Strength is not one of them.

Second Answer -- If it IS Wild-Clasped/Wilding, YES!

So, the fact is that because you retain your type when Wild-Shaped, literally anything that worked for you while in normal form (aside from things you literally could not use like a dagger or something) will also work on your Wild-Shape.

This is why wilding-clasps are essentially item-taxes for Druids, and also why the MIC rules for combining items is such a HUGE deal for Druids, because it maximizes the amount of stuff they can cram into a few slots that will be wild-clasped.

If you Wild-Clasp something, though, the general answer is that it will absolutely work in Wild-Shape!

As a bit of an addendum...

The SRD tells us that items you cannot use in Wild-Shape are either melded or dropped. If you assume a form that is humanoid...or a lot like a humanoid, chances are your items will be wearable.

The SRD also states that magic items can change size, so that's not a problem, thankfully.

In general, just think about the form, and use some logic. I love to play Planar Shepherds, which means most of my time is spent in Outsider forms. Now, some of these are really, really impossible to wear anything in, but most Outsiders are humanoid. Right now in a campaign I'm Wild-Shaped into a Kelvezu (type of demon).

In this case, my armor and cloak wouldn't be wearable because of my wings, but everything else would be wearable because the rest of the body isn't notably different from a Human (my character race). This means that some of the stuff I don't have wild-clasped, like my rod of lesser extension, are still hanging on my belt ready to be used. My armor and cloak are both wild-clasped, so they still work despite my wings.

If I were to take another favorite form, that of the Elder Viper Tree, then EVERYTHING would have to be melded unless it was wild-clasped/wilding because trees simply don't have shoulder slots, head slots, etc.

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.