I'd say the triac is part of a soft start circuit to reduce current inrush at turn on.
This being shorted could blow the fuse, however if it works I suspect something else is up.
You could check the diac, the device inline with the triacs gate, with a led and 10k all in series with it, the device should switch on at about 20v or so, meter it first if its shorted that could be yer problem.
A lot of times on old 'scopes the ht supply to the tube is the issue, dampness gets in and messes things up.