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Just an FYI for anyone curious.
If you're using Windows 10 and have the Xbox app (whether you have an Xbox or not), you can enable the game DVR (default WIN+G). When running Remote Play, use that combo to open the game bar, tick "Yes, this is a game", and voila. This will allow you to record gameplay without needing the Share button.
You can also use GeForce Experience, by adding the directory where you've installed Remote Play into GFE. This seems hit and miss; it's worked for me more often than not, but it also fails to register sometimes.
In addition, you can use most other recording software (OBS, FRAPS, etc.)