Before replacing caps, I usually do some preliminary inspection, cleanup, and tube testing/cleaning:
http://www.antiqueradio.org/FirstStepsInRestoration.htm
The following page lists articles about restoring various old B/W sets. None of them is exactly like yours, but if you skim a couple of the B/W articles, that should give you an idea what's usually involved in getting an old TV to work reliably:
http://www.antiqueradio.org/restoration.htm
Some plastic flaking off a coil doesn't seem like a big problem. If you have never restored a tube radio or TV before, consider practicing on a simple tube radio before tackling a TV.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html