shouldn't the smart tv already have youtube on it? not seen one that does not have it built in. you should not need the phone to make this happen.
A lot of TV manufacturers don't bother updating the apps on their smart TVs. For YouTube in particular, over the years Google has disabled Flash and made VP9 the default format. And a few years back they changed how the videos stream (it now comes in chunks) to thwart people who were capturing the streams to save videos locally. So smart TVs with un-updated YouTube apps can no longer play YouTube videos.
This is yet another reason you should generally avoid smart TVs, and rely on a standalone streaming box to add "smartness" to your TV. The box will be much better supported in the future. (The other big reason is that updating the firmware on a smart TV can actually reduce the number of features it has.)