> As far as I know it is Cloud integration which enables this absolutely necessary and table-stakes functionality. Is that not true? Does e.g. Keepass provide this essential functionality without a Cloud integration of some kind?

Just store it in your regular sync solution. Syncthing works great, and I don't remember any issues with Dropbox back when I used that. I'd imagine that iCloud or SkyDrive would work fine too, for the masochistically inclined.