I'll bet youre barking up the wrong tree with the vertical. The horizontal is weak too, and probably the high voltage... from that picture you posted it looks like everything is weak.
I think the other posters had it nailed, its a low-voltage power supply problem. All Zenith roundies (as far as I know) have a voltage doubler power supply. Its the same circuit used in RCA ctc9 to 15, Silvertone, Philco, even Setchell-Carlson! (havent seen that one, only the schematic). It makes about 400 volts.
As I understand it it works like this, theres 2 diodes and 2 capacitors (160mf, 250v). When the current from the power transformer flows one way, it flows through one diode, and charges one capacitor (to 200v or so). When it flows the other way, it flows through the other diode and charges the other capacitor (also to about 200v). Because the capacitors are in series, you have about 400v output to the filters.
You can check this, just measure the DC voltage across each 160mf 250v capacitor with the set running. Each one should have about 200v. I'll bet one is really low. When you find the low one, substitute another capacitor (unsolder the old one). If the capacitor doesn't fix it, you probably have a bad diode.
Unsolder one side of a diode. Hold onto the lead with a needle nose pliers (diodes dont like heat). When you have one end loose, check it with an ohmmeter. It should measure really low one direction, and really high the other. If the first one is good, try the other.
Of course I might be completely wrong about this, but if both halves of the voltage doubler are ok, start going down the power supply chain, checking the voltages against the sams.
Gorgeous set
Hope you get it fixed.
John
P.S did you mean in that last post that the old caps are still in the circuit along with the new ones? I wouldnt do that except for a quick test.