[Fred]

All above sounds spot on.
Does sound like float needle valve is stuck in the open position, perhaps ever since the mechanic "rebuilt the carb" and issue then was only revealed when petcock was opened for your ride.
Having someone do the work is nice and convenient only if you can trust them to do a better job than you can. Can be hard to find someone familiar with TWs though.
Disassembling and inspecting/ replacing the float's needle valve is not difficult and makes for a great learning experience. The more you understand your TW the more you may appreciate these litle machines.
Maybe I'm goofy, old fashioned or just curious about the world I live in but I feel the need to understand any system I am interacting with. Since bad things can happen far from home this understanding seems especially true for a motorcycle.

After all you don't want to be like this guy I came across middle of nowhere Nevada sitting amongst all his worldly possessions next to his blown Triumph. Not easy to find only available shade uncer the bike when you have to share it with a growing oil spill. Unfortunately he knew almost nothing about bikes, was moving from east coast to west coast and new no-one in between other than this strange guy with a Betty Boop decal on his bike who stopped to help.

 

[Fred]

R.E. the broken down Triumph: I didn't do anything any of us wouldn't have done to help a fellow motorcyclist.
I stopped, gave him shade, cold beverages and my cell. He called GEICO who was very helpful and we worked to get him a tow to the nearest Triumph Dealer to the west, tried unsuccessfully to get him towed all the way to Bend Oregon. Hardest part was my explaining to operator that the nearest cross street she needed to complete the paperwork was a good twenty miles or so away, and had her put in an estimated mile post reading. I do not recall another car going by in the twenty plus minutes I was there before resuming my eastward journey to Moab.
Point is had the young man understood his machine better he may have detected a developing problem before sufering catastrophic failure far from help.