WebDAV arguably has fewer essential protocol level issues (aside from being overly large and baroque) -- that is, it has fewer problems that are necessarily included in a correct implementation -- but lots of implementation issues (which, arguably, stem largely from the spec being overly large and baroque.)
OTOH, even to the extent that the problems (e.g., with WebDAV) are issues with the existing software and not the protocol, the main benefit of building a specialized app to a standard protocol is interoperability with other tools using the protocol -- if the existing implementations are incomplete and buggy in mutually incompatible ways, that negates the benefit of writing to the standard.
With most FTP server software I know (such as ServU or FileZilla Server) once you force SSL connections, the server will disconnect users on the USER command (before sending the password) if they are connected insecurely. So only if your system administrator wants to allow insecure connections, will you be allowed to transmit the password insecurely.
Why do people still connect using the old insecure FTP standard? It's the software.
I personally think FileZilla should connect using SSL Explicit by default. IQBox defaults to SSL, and prompts the user for a fallback. So again I think it's a software issue, not a protocol issue.
WebDAV is terrible. Run, don't walk, away from it. Nobody at apache has any responsibility for it, the mod_dav author is totally MIA ... it's just a mess. Further, every single DAV implementation[1] is both broken and completely unique.
[1] Like DAV in Finder, or DAV in MS Word or in MS explorer, or in IE ... they are all completely different ... and also all of them are horribly broken.