Went back to the top listened to the start run video a few times,
thoughts.
we haven't seen the whole bike, carbs, air filters, exhaust etc.
Video sound is not the greatest, but my 1000 mile diagnosis is that it sounds like it's going lean.
idle sounds great.
your pickup test seems like they are fine.
can you "horse the RPM up" with choke or cadging the throttle? Are both intake air barbs capped? might remove the vacuum line from the LH carb, cap and fire with petcock in prime.
Are there "extra fuel filters? Any rust in tank?
try to get it past 2500 rpm, ideally let it stall trying, pull the plugs, what do you see?
open the float bowl drains run a line to a clean container turn petcock to prime watch fuel flow. look for good flow, crud. Then it's into the carbs.
From the carb tech thread
http://www.xs650.com/threads/carbs-carburetors.43/
A thread devoted specifically to BS34's as found on the XS650 80-84
http://www.xs650.com/threads/carburettors-from-the-grave.34737
The BS34's have a small "final" fuel screen inside the carb body above the float valve seat!
Rather obscure but exhaust blockage? there are occasional instances of a collapsed stock inner exhaust pipe wall, to be honest I no longer remember what years this happened on.