How to change the browser of a web-shortcut?
Try Website Shortcut. Open the app, paste the link of the website/webpage, then click on create shortcut. It will fetch the icon from the website for you.

On the home screen, click on the shortcut, you will be prompted to choose a browser. After selecting your browser, I advise to select "Just once" to allow the selection of other browsers if needed.

The disadvantage of the app is that if you remove it, the shortcuts will also be removed.

Disclaimer: I am not related to the app's developer.

EDIT: Below his the response I received from the app's developer as to why the source code is unavailable. He has given his authorisation to show the screenshot of his response.