Hey, great minds think alike! I just finished restoring one of those this week.
The horizontal output tube seems like a good place to start. On mine, that tube had gone to air, and I noticed that it was a replacement Sylvania, not an original RCA tube. So, my set had gone through at least two of them, although the original CRT is still very strong. Maybe it's hard on HOTs, or some other dodgy component stresses them. Of course, everything gets hot inside that tight cabinet.
This is the second 14-S-7070G that I've owned. The first was sold to another collector years ago, but I missed that classy red & gold set, so I grabbed this one when it showed up cheap on the local CL.
I did replace several resistors in mine, including one that had gone completely open, disabling the brightness control.
Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html