It's not uncommon for B+ to be a little high, since our line voltages are higher than the 117vac specified in manuals of that day. I wouldn't worry about it for the time being. Longer term, if it bothers you, you can run the set from a variac or humbucking transformer to reduce the line voltage.
Cleaning the tuner and tube sockets is a good idea. Before that, I'd clean the volume control and other potentiometers. Dirty pots are almost universal in TVs of that vintage, and sometimes a simple cleaning can work wonders.
If the control case has a little opening by the terminals, you can spritz DeOxit into the opening, then work the control all the way back and forth a number of times. Let the cleaner dry before powering the set again. Some controls don't have an opening; you can open those by carefully prying up little metal tabs on the case. Don't bend the tabs too hard or they may break off. This is easier to do with the control removed or at least loosened from its mounting nut and moved back from the chassis.
I clean tube pins by gently brightening them up with very fine sandpaper. After the pins are clean, you can clean the socket (at least somewhat) by spritzing a little DeOxit into the socket holes and/or on the pins, then plugging and unplugging the tube a number of times. Take out the tube and allow enough time for all the cleaner to evaporate. If you do one tube at a time, you won't risk mixing them up.
The contacts on tuners can become oxidized, like any switch contact or control surface. Do not use sandpaper or anything abrasive to clean tuner contacts. The lazy man's cleaning method is to spritz DeOxit on the contacts and work the tuner through all of its positions several times. A better method is to dip a Q-tip in a little DeOxit and rub each contact surface. After cleaning, you can go back over every contact with an electronic lubricant (also made by CAIG).
If you're getting reception on a couple of channels, I'd go ahead and install the picture tube to see what you get. Once you have confirmed that your CRT and flyback are good, you will have time to investigate other issues such as no reception on a certain channel.
The 630TS is one of my favorite TVs. This article has a link to the complete RCA service manual:
http://antiqueradio.org/RCA630TSTelevision.htm
Phil Nelson
http://antiqueradio.org/index.html