This will probably break many other use cases as well (e.g. services that allow pasting formatted text and will convert it to unformatted text, or pages like GitHub where you can paste an image into the edit area).
The only service I am aware of that it does break is Google Docs, because they handle things in an odd manner. But as they received several bug reports for that since 2017, I wouldn't get your hopes up that it'll ever be fixed.