Shooting from the hip I'm going to guess you have a B+ supply issue, likely one of the larger-wattage resistors in the filter network opened up. Also if the 'lytics are original then you need to seriously consider replacing them--especially if they are showing any kind of spittle around the terminals--or are running hot to the touch....
If your horizontal oscillator isn't working then your horizontal output tube will climb up in current and kill itself. The bias it needs to keep the cathode current in check is developed by the driving signal from the preceding stage.
I would recommend you try and locate the Sams folder on the set before doing a lot in the way of troubleshooting; Zenith used point-to-point circuitry, and--while it is superior IMHO to PC construction in tube sets--it also is the very dickens to read, as component density tends to run high.
I had an early '60s Zenith roundie which I enjoyed for years. One quirk it did have however was that it ate the focus rectifier every 18 months like clockwork. Go figure....
