This is probably a protection circuit kicking in especially if turning power
off or pulling the plug is required to restore operation.
The detection circuit could be in the power supply or horizontal deflection
output circuit. It may be defective or the current may be too high for some
other reason. A couple of tests can be performed to confirm that it is due
to beam current:
Determine if behavior is similar when adjusting the user brightness control
and the screen (G2) pot (on the flyback) or master brightness control. If
the TV quits at about the same brightness level, overcurrent protection is
likely.
Disconnect the filaments to the CRT (unsolder a pin on the CRT socket) and
see if it still shuts down under the same conditions. If it is overcurrent
protection, shut down should now NOT take place since there is no beam
current.