I believe we originally left it this way so as to keep api.php as "normal" as possible within its already unusual use of the Skin system (we might handle this differently if we did it again today). So allowing experimentation and innovation through gadgets and user scripts seems like we might as well.

I recognise that it is not uncommon for interface admins to forget about non-default skins, but I don't know if that's reason enough by itself. For one, I would expect these to fail without visual problems given ResourceLoader's isolated execution model. And beyond that, sites have control over these the same as way for any other skins. E.g. if a gadget isn't compatible or intended for use in Timeless, its definition can be set to something like skins=vector,minerva,vector22. Likewise, common site scripts intended for specific skins only can also use MediaWiki:<Skin>.js instead of Common.js (or use a default/hidden gadget like I did on mediawiki.org).

Another example of commonly broken scripts is ContentTranslation (Special:ContentTranslation) which has also begun using the Skin system (overview of up/downsides at T291193#7360586).

Having said that, I don't really mind either way, and at this time I'm not (yet) aware of anyone using this for good. We could ask Wikitech-ambassadors if we want to know if anyone's making use of this.

(Just to mention the actual skin name implied by your comment, it’s apioutput apparently.)