I’ve taken a closer look. The service you reference in the link above is a service for adding link buttons to web sites. The chrome extension I think is this one, and is effectively acting as a middle man in Chrome to allow you to have the equivalent of those buttons on a site without adding them to the site.
The AddToAny service looks to be exclusively designed to integrate with online services. That’s not a huge surprise, and seems logical. However, what you seem to be asking for is to integrate with an application that isn’t an online service using this AddtoAny service. That means you’re trying to do two entirely disconnected things, like trying to get a electric car to run on petrol/gasoline. You can do it, but you have to build something to turn that petrol /gasoline into electricity and pipe it into the car.
So the question of viability comes down to whether or not you can build a bridge from an online service to local app instance. If I had a personal need to, I would absolutely attempt the latter that I described above. I have a Mac mini on 24x7 that I could use to facilitate it.
Looking at the capabilities of AddtoAny, it doesn’t work with any service, just the services it has created links for. As a result I’d probably look at something like using Pocket as a go between to receive the links, then something on the Mac to process them and push them into Drafts via the iCloud import folder.
However, if Chrome wasn’t giving me the native share options I would expect (I don’t use Chrome on the Mac, but I assume that is the underlying issue you are having), I would use a different local method. The simplest would seem to be to copy the content to the clipboard and then use Draft’s global hotkey for capturing the clipboard. You could streamline that further with something like Keyboard Maestro to trigger all of this and clear the last entry from the clipboard history if you wanted to keep that unlittered from such sharing options.