Thanks for the suggestion. I already did that before posting here, and it does not resolve this issue. However, the options are called:
Steam > Settings > Downloads > Clear Cache
Steam > Settings > In Game > Delete Web Browser Data
I did both of these. The first option restarts the client, but the second does not (you have to restart manually).
Edit: I also tried disabling these features, and neither of these fix the client issue either - Enable GPU accelerated rendering in web views, Enable hardware video decoding.
Note that the issue in the steam client looks exactly like the issue you post above. Not only does the video not play, but there aren't even any controls for the video that can be clicked on. I'm on Windows 10.