I think the kind of plastic used in bottles is one of very few that actually are profitable to recycle. PET, I think. This is actually something recycling companies want. Most other plastic is just burned or shipped somewhere.

IIRC the reason most caps used to be unrecyclable (many still are) is that they had a liner in them made of a different material. Because such caps were composite materials (using different types of plastic for the liner and the cap), they would make an impure product if recycled. The same problem applies if the cap and bottle are different types of plastic, which used to be more common.

(By "recycled" I mean most likely shipped to Southeast Asia to then most likely just find their way into the ocean)

You would be wrong; PET bottles are mostly actually recycled, because it recycles very well (also, why would places go to the effort of setting up a deposit/return scheme for something they aren't actually recycling? Just throw it in the blue bag with all the other plastic that doesn't actually get recycled)

I remember they used to collect bottle caps to donate to charities that used it to make stuff for dogs and stuff

This is why I stopped recycling and put everything in the trash.

For one I was so sad when I found out what a scam it is that such a small amount of stuff is actually recycled and especially plastic never was actually planned to be recycled.

Two, I'd rather most of the plastic end up in our own landfill than to be shipped overseas into someone else's or just end up in the ocean. We deserve to poison our own land thanks to our politicians not holding corporations to account.