that's basically how it already works...

extensions choose on which site they're active and if they provide any available assets (e.g. some extensions modify CSS of the website by injecting their CSS, so that asset is public and then any website where the extension is active can call fetch("chrome-extension://<extension_id>/whatever/file/needed.css" if it knows the extension ID (fixed for each extension) and the file path to such asset... if the fetch result is 404, it can assume the extension is not installed, if the result is 200 it can assume the extension is installed.

This is what LinkedIn is doing... they have their own database of extension IDs and a known working file path, and they are just calling these fetches... they have been doing it for years, I've noticed it a few years back when I was developing a chrome extension which also worked with LinkedIn, but back then it was less than 100 extensions scanned, so I just assumed they want to detect specific extensions which break their site or their terms of use... now it's apparently 6000+ extensions...