I was a happy user of that workflow until I started working for an organization that blocked Dropbox but not any of the browser plugin based password managers.

Also while free, arguably the UX is not very good especially on mobile, unless Keepass integrates the way Lastpass, 1Password, et al do. I cannot imagine convincing any of my non-tech friends to go this route.

Interestingly this is basically how 1Password did password sync for years - not a Keepass database, but a 1Password folder structure stored within Dropbox saving a bunch of little text files. They added other synced storage options over time before turning up their own cloud service, but third party sync was where they started.