I would second Bill's advice to replace all of the electrolytic and paper capacitors. After 60 years, they are all bad, and it's pointless to try detailed troubleshooting until they are gone.
In my 630TS, I also replaced four mica coupling capacitors and a few resistors. This article provides details.
http://antiqueradio.org/RCA630TSTelevision.htm
Have fun! The 630TS is one of the best designs of its time and performs well when restored.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html