I don't watch "Pawn Stars" regularly, but have caught the show in the past. The "gun expert" they had on the show is pretty good with WBTS and later arms, but not so good on earlier arms. That is not meant to be a hard criticism of him, as no one can be truly good in all time periods of all guns.
Having said that, I did not feel comfortable with him pronouncing it authentic just by patina/condition alone. I wish the lock engravings and the barrel stamps were visible in the video.
I did notice some things wrong about the musket even with so short of a time to look at it in the video.
The top jaw screw has a hole in it and that is NOT correct for a 1762 lock. Top jaw screws with a hole did not come out on British locks until the 1770's. That top jaw could have been a replacement during the AWI or later, but it still makes the lock suspect.
Also, the screw heads of the Feather Spring screw and Steel (frizzen) screw look as if they were at least cleaned or polished and may be replacements as well. The Steel (frizzen) screw looks like it may have a parallel cut turnscrew (screwdriver) slot, which did not come about until the early 19th century at the earliest.
Personally, there is NO way I would pronounce that musket authentic without AT LEAST pulling the lock and examining the inside of the lock, the lock screws and the stock mortise for the lock.
Gus