I too work at a company/in an industry where FTP (incl. sftp/ftps) is still crucial, for both inbound and outbound transfers. A great many of our partners would find using other protocols more cumbersome. Automating the download/upload of entire directories of arbitrarily named files seems considerably easier/to require less code than any solution over HTTP might (log in, perhaps change directory, `prompt`, then `mget/mput *`). Albeit I've not done much research into alternative means recently, but a dedicated protocol beats a hand rolled solution and anyway inertia is strong - especially as (for us) nothing is transferred in either direction that isn't destined to be made public by one party.
This is tangential to whether support for FTP is still worth keeping in a web browser, mind.
