Look for the app with the old TV set look. It may have moved accidentally to the second screen so swipe left to have a look there. Or, maybe restrictions are turned on and YouTube is hidden - check in Settings. I'm not sure but there may be restrictions on YouTube in certain countries too. Whereabouts are you? It's a standard App so cannot be deleted.
If you type in "YouTube download" into the AppStore search bar you will get a number of downloading apps that claim to be able to download YouTube clips.........I can't vouch for any of them as I haven't used them. The Archangel
Tagdisk does a good job of saving Youtube videos to your iPad. The Lite version is free and only lets dowload up to five clips I think, but easy to use and works perfectly
I have the paid version and many od the videos that I chose to download never showed up in the downloads. Sent from my IPAD 1st generation. WIFI only. 32 GB