Is there a reasonably straightforward way to download videos from proprietary web browser players?
'Easier' solution: have you tried Firefox on Android and the myriad of video downloader add-ons you can get there? One of them might work. I don't have the time to test them all now but let us know if one works.

'Harder' solution: I tried digging around on that site you gave, on a desktop you just need to fire up dev tools and look for the first .bin file in the network tab (once you start playing the video) then throw it into youtube-dl [url]. Tip: use the search filter to find it. I feel there might be some way to automate this with a Python script or similar. 

https://rg3.github.io/youtube-dl/