A capacitor can leak internally when voltage is applied, yet test as "not shorted" using an ohmmeter. The voltage applied by a general-purpose multimeter is negligible compared to the actual working voltage. That's why they made dedicated capacitor testers. Mine can check a cap at any voltage up to 500v. I haven't heard of any cap checkers that can test in the KV range, however. For a doorknob like this, the test is substituting a known-good cap and observing what happens. Which was no problem in Ye Olden Dayes when the repairman would just fish one out of a bin on the shelf and pop it in.
If it looks like you need a new doorknob, Surplus Sales of Nebraska has one at 500pf/20KV for a low, low price of $35:
http://www.surplussales.com/Capacito...cap_trans.html
Click on the "Available in these styles" link to see whether that type would replace what you have.
It's not a cheap part, so I'd proceed with whatever testing is needed to isolate that as your trouble source.
TVs can have more than problem at a time, of course. If you also want to replace the 1X2 filament loop, I believe that needs to have HV-rated insulation.
The photo shows some sensor-safe RTV that I used on a Philco flyback.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html