The most tedious task I do with LibreOffice Writer is cleaning up badly formatted documents. To do so I use a "trick" or "workaround" which is described here:
I would suggest to implement this inside of Writer (other programs in suite would benefit as well). Even better, something like this could be the default filter when pasting clipboard text from unknown HTML sources.
Here is a simple text-mockup showing how the dialog could look like:
[Clear direct formatting] button
[ ] Underlining
[ ] Webdings fonts
[ ] Remove all whole-paragraph styling
[ ] Ask when in doubt
"All bad formatting is equal, but some is less equal than other" ;)
But I think your idea is useful.
[Side note: From what I remember, I think at some version applying a different paragraph style would not remove direct formatting of words, and now it does - or the other way round.. should just check of course.]