I came here looking for more information on this. What I can add to the discussion is this; when I have already initialized a stream remotely with my Nexus 7 (gen 2) I am immediately able to see the active stream being sent the the chromecast from my iphone6. Trying to start a brand new stream with my iphone however has the same issue described by OP.
I'm betting this is by design and not a full featured app like the android version, which would be a poor decision if it's avoidable. Myself & my wife both have iphones and the only remote device I have that works for starting a stream is my android tablet which is not always at home when I'm not there.