Pizza boxes are technically recyclable, it’s just that they have to be recycled differently to other recyclable products due to the food particulates left behind on the boxes, and also other recyclable products cannot be contaminated by the same food particulates, hence why they likely have a separate section for pizza boxes.

That rule varies by location. Some facilities can handle pizza boxes, but not all of them can. It’s best to call your local facility to make sure.

I find this hard to believe, but I've also done zero research. I have, however, dropped off many old pizza boxes at various places, and they always accept them for recycling. Maybe it's just more expensive and labor-intensive, but I doubt it's impossible...

That's a pretty old rule. Last decade or so most recycling programs support and encourage pizza box recycling. Check with your local center.

And they literally put it in the box, telling me to recycle it.

Maybe the special slot allows them easy separation and then they burn it.