And that is what is happening. You tube URLs in safari are opening in YouTube app.
What is being requested in this thread is to watch the YouTube video in Safari, aka letting the user pick which app to handle the link.
No, if you read my posts, you'll see I pointed out that YouTube is not conforming to the nice method of universal linking introduced in iOS 9. I posted an example of Twitter, which does it nicely, where the OS provides a link in the top right to go to the web page view instead of the app view. YouTube does not support this feature. Another annoying feature of YouTube's method is that it pops up a box (where your options are only to cancel or go to the app) whereas Twitter just goes straight to the app, but then lets you go to the web page with the top right button.
If YouTube followed Twitter's lead (and Twitter is just one example here. ebay do the same, to give one more example) then OP would have what they wanted
edit:
If anyone doesn't know much about this new Universal Links feature I've done a bit of further research into it
When an app supports this feature and a URL to its website opens the app, as you know you can then tap on the top right of the status bar to go to the website instead. iOS will take that to mean you always want to open the website instead of the app. You can revert to opening in the app by long-pressing a link for that app's website and choosing Open in "App Name" - after that, the default tap action on its URLs will be to open in the app. When the default action is to open the app, you can also use the long-press menu and chose Open to open the website
I've added some pictures:
1. Finding MacRumors twitter profile in Google search
2. Default result of tapping the link - Twitter app opens on the MacRumors twitter profile
3. Tapping the top right status bar button which takes me to the website of the MacRumors twitter profile
4. Showing the long-press menu on the Google search link to show both Open and Open in "Twitter" options