Adobe Illustrator images in Adobe Stock have checkered backgrounds

Those assets work only against a background layer. The chequered background, as you correctly mention, is a background layer or element and does simulate only a transparent background, as otherwise you would not see the effect.

See here for a similar case: https://community.adobe.com/t5/stock-discussions/transparent-png-from-adobe-stock-not-transparent/td-p/12526859

(Please note that this post is from 2021. As the core of the answer is still true, Adobe now offers PNG files.)

You need to insert a background into a separate layer and then you sort of have to modify the other objects to match the colours. Just needs a bit of effort and know how.  Some experience and patience is required.

These are true vectors and have to be designed this way so that they are truly scalable to any size. Those "circles" are just necessary evils I guess.  As I said earlier you just have to fiddle with the colours and possible gradients of those "circles" until they look right with your BGs. In other words till they no longer look like odd circles.

I hope that helps somewhat?