In windows you can choose what application opens which file extension (duh). For instance, a common one is setting Adobe Reader to be the default program to open .pdf files.
Edge (or most chromium browsers) have also a functionality where some web applications can be "installed" as pseudo-standalone applications (see example tutorial)
I used this functionality for https://app.diagrams.net/, which saves diagrams with .drawio extension. However if I want to open a .drawio file, I must open it from the app itself, I can't double click the file and have it opened by this "web app".
Is there a way to associate this extension with this web app so it opens automagically?
I think your requirement can not be achieved.
After testing, I found that you can't even find the drawio file type in windows default apps setting. Therefore, such drawio file cannot be recognized by default, and only after drawio-desktop is installed, it can be opened in this application.
I also tried the create shortcute you mentioned, but it also doesn't work. Edge/chrome can only parse it as XML data and display it on the page. So I suggest you can use drawio-desktop to open/process it as I mentioned above.
You can also create a shortcute of the diagrams.net
and drag the drawio file to the page when opening this page.