Surf with chrome logged in? Will never happen. Hard no as long as my sanity remains.
Firefox, maybe, haven't looked into it and just the process of evaluating it isn't worth it. Mozilla doesn't have the track record that I want to blindly trust them with that either.
As search engines have gotten better over the years, I find my use of bookmarks is usually write-once read-never.
Part of that is due to the fact that iOS is now my primary consumption platform, and finding bookmarks in a complex structure is much more awkward than it should be.
Part of it is just laziness: trying to manage a complex bookmark folder is more work than I care to do when search engines give me 99% of what I need.
For what it's worth I had the same problem with folder/hierarchically structured bookmarks, but tag-based bookmarking (like del.icio.us used to provide) meshed very well with me and my workflow so I wrote https://savecrate.com (still a WIP and rough around the edges but functional enough to work for me)
Chrome has no other export functionality. And firefox copied that lack-of-feature without even realizing the motivations.
So now one have to copy them one by one.