I'm wondering if there is a better way other than asking users to point a camera at their TV screen and try out some tasks while recording. Maybe mirror the TV screen to a computer and capture the computer screen?
Yes it's probably better for you if the user can mirror the TV screen into a computer. However, it's a setup that might require some extra tech knowledge or perhaps extra hardware (maybe the user has an old computer without HDMI ports? maybe they don't have the software to capture the screen? maybe they don't know how to set it up?).
You want to keep this simple for the user, so they can focus on the actual test. For this purpose the camera pointing to the TV is better: nowadays mostly anyone has a phone capable of recording video.