You are right, ICE does support TCP. Not all ICE implementations support it though, probably where the incorrect anecdote comes from.
ICE TCP did get a little more limited recently in Chromium though [0] because of the TCP port scanning issue [1]
I have also heard that some gateways/firewalls/$x don't allow any non-TLS traffic, so you can't even establish ICE. In those cases the DTLS/TLS transport of TURN is nice.