Over time, the cabinet hinge has progressively become looser, and there are noticeable cracks developing in the wooden component. I'm interested in finding out if there's a permanent solution for stabilizing the hinge with the cabinet door or if there's an alternative approach to resolve this problem.
I'd go for a 2-in-1 fix.
You need to strengthen the cabinet frame, fill all the existing holes & glue the split section. Then I'd go with Solar Mike's idea of adding a batten on the inside, firmly screwed & glued. This will not only add strength to the existing structure, but provide a basis for part 2…
Replace the hinges. The door insert is a fairly standard construction, so that part will be easy to match, but for the cupboard carcass itself, you now have more thickness to play with, so you can use more standard fittings.
Something like this, which is not only soft-close, for less door-slamming, but also can open to 165° so they won't hit a hard-stop when someone opens the door.
Get these fastened into your new batten [they're a longer throw than your existing ones so that should be fairly easy] & you'll have increased the strength considerably.
Rather than trying to choose exactly the right hinge online, take one hinge & your pictures to a local merchant & get their advice.
I’d remove everything, inject a thin glue into the crack and screw hole (ie, “chair doctor”) and clamp until it dries.
Then, find a screw that’s a little longer than the original (if the original was an inch, go with 1-5/8”), predrill (drill bit should be a tiny bit smaller than the root diameter of the screw), and reinstall.
If you have kids in the house, ask them nicely not to push the doors past their normal opening range. These cracks happened for a reason.