Yes, it can be done. It is a little bit of work to setup properly, but once it’s setup properly, it works good. I am using it for a couple of my most frequently used tables, but you can set it up in such a way that one setup can be setup to be used in different places of your doc and applied to different tables, for each table with its own filter settings.
It has been worked out and documented by Paul Danyliuk and you can find the posting here:
It has been a while since I looked at the video, so I can’t remember how detailed the explanation is, but once I figured it out, I never looked back. The only drawback (in my opinion) is that you use a field in a table to enter your search terms. By hiding columns, title, headers and the new row symbol the end result is pretty clean, but I hope for a better tool in the future. The newly introduced text object would make for a nice search field, but for now it has the exact samen problem you are having now.
The other drawback is that you have to think in advance on which columns you want to filter, because if you use the build in table filter options, you end up forcing your filter on the other users.