It appears the part you're pointing out is the rectifier connector. The other end would connect to the stator connector part 2.

I would suspect it's burnt as the connectors seem to be too small to carry the load being produced and over time the connectors melt. At least this is the case with my (and many other) 70s vintage Yamahas. I would suggest if you can access it, replace the OEM connector with spade connectors.

I don't see much wrong with the voltage relative the RPM unless it's too low at max RPM. In the beginning of the video at regular idle, the bike makes just over 12v. At the end before it dies the idle is too low for whatever reason so the volts produced at that lowered RPM is under 12.

Would suggest you spend some time at ViragoHelp

Where I got this bit:

"So when our engines are revving at 3000 rpm, more current is going to be generated than at say 1000 rpm at idle. The fact is that at idle not enough current and voltage will be generated to meet the needs of the electrical system, plus keep the battery charged."

and

"If you suspect a charging problem your first move should be to grab your trusty multimeter and take a voltage reading across your battery terminals with the engine running. The manual will tell you should see 14.3-15.3 volts at 2000+ rpm. (14.5 for earlier models). In my experience these voltages will vary a bit , depending on the state-of-charge of your battery and other factors."

Of course there are many varieties of Virago so you might want to spend some time on a bike specific forum.