In such cases with live streams, there you are better off with a Shield TV Pro, where you have the ability to become a developer, so you can force HDCP 1.4 on everything.
Then you need a HDMI splitter that can remove the HDCP 1.4 and then get a proper HDMI recorder that also can handle AC3 and E-AC3 to record the eventual 5.1 audio as well, where you then use your computer as a stream recorder.
The recording hardware does not come cheap, so its also mostly only sponsored professional online gamers who are using such HDMI hardware.