Hoping I get the problem right - you just want to search quickly YouTube in the open tab when you're in safari right?
In a personal manner, I switched my search engine to DuckDuckGo (can be done in the Safari settings) for privacy reasons, but also because of their "bangs!". If you search "XYZ" you'll get DuckDuckGo's results. "yt! XYZ" will search for XYZ on YouTube, "i! XYZ" searches for images, "g! XYZ" searches on google etc.
So if you hit ⌘⌥F on your keyboard you'll be able to type a new URL in your current safari tab. Now in BTT you could make a shortcut like ⌘⌥⇧F that first makes ⌘⌥F and then pastes "yt! ".
Hope that helps anyhow
Thanks. There was a Safari extension where you could press "y" then space in the search bar in a new tab and your search and it opens Youtube. I'm looking for something similar.
Andreas suggestion comes closer to what I'm looking for BUT it opens a new window(not in the same tab) every time I search.
I believe the problem is that if someone uses the Safari search bar in a new tab BTT's default settings is to open a new Window.