I retested this today after rebooting the TV and phone. I got better results, but the user experience is still a bit variable.
The "JRemote2 cast scene" seems to just be a black screen with the words "JRiver JRemote" in the centre. That's fine. I know casting is on from that.
Yesterday the movie I was playing didn't appear in Playing Now, but today it did. So that was probably as a result of rebooting the TV. Or because I had played something locally on the phone prior to testing.
Tapping vs pressing and holding on a movie gives different results. Tapping was a bit inconsistent, but when it worked all movies in the View were added to Playing Now (well, a lot, I didn't check exactly which, just that the one I tapped was next to play). This behaviour is more suited to audio than video, but that is fine. Pressing and holding, then selecting "Play Now" added only the one movie to Playing Now, which is much preferred for movies.
The screens shown on the phone while a movie was playing seemed to vary depending on the format of the file being played. MP4 and AVI files were different to TS files. I was using the Low Quality video setting initially, and the High Quality setting later. Same results.
The movies still paused after a minute or so, with the spinning icon, and didn't recover.
I guess Chromecast will become more reliable when you change the streaming method.