Look in the settings of the YouTube app on the TV and see if you can find a place to just sign in. You shouldn’t have to have it paired with the phone just to sign in. And, you might find if you can sign in, that you see the TV in your phone app YT settings. At least, though, you can search for things on your phone and find the search on the TV app (at least if it works like other YT apps). If not, I hate to say it, but it might be worth adding a streaming device. Roku has some inexpensive options and works with almost all content (they are “content agnostic”), whereas Amazon’s FireTV sticks will always push Amazon content first. Roku is probably the best option for inexpensive streaming. And both of these have more streaming options than something like a Chromecast (which is great, but requires the use of a phone/tablet/computer to actually “cast” the content from).