OK, so I can add native bookmark to a browser, but then I may be on another machine or browser and want to open that bookmark.
My suggestion is not to have each browsers native bookmarks sync with each other, more to have the app keep a section (say chrome/edge/browsernamehere) that is a regular backup of the bookmarks native to the browser.
this way if I bookmark locally on the bookmarks bar on one browser, I can pick it up from the app on another.
This removes the need to do back and forth sync.
just once every 20 minutes or so, have the app slurp the browsers bookmarks