Something really wrong with the measuring.
First -- Z1 pin 14 and Z2 pin 14 are connected to each other. One having proper power but the other not points to a broken trace.
However U1 and U2 showing No power at all indicates the meter's power lead was probably to a wrong place or not making good connection.
Try these again but make sure the meter is well grounded.

As far as C3 and C4 measurements - you measured these in circuit so you are measuring capacitance of other 'things' mounted in parallel. Can't reliably measure these in circuit.

Measure voltages across C1 and C3 to see if -45 and -15 are ok.

There is definitely a broken trace or bad solder joint in there somewhere. Z1 pin 14 needs 5V at pin 14.
C6 is Z1's bypass capacitor -- C6 has good power, Z1 does not. Follow trace from C6 to Z1. Are you measuring this at the actual IC pin? If so, could be a bad connection between pin 14 and an IC socket (if Z1 is socketed).

Most cap voltages look good but there is an oddity across C2.
C2 should read 7V from plus end of capacitor to minus end of capacitor but was probably measured from ground to ???. What voltage do you get across R2? This will tell us the offset voltage going back to your transformer.

Much better.
The voltage measurements all look good. A tad low for the 5V but shouldn't be an issue.

Voltage across C1 should be same as voltage across VR1 plus voltage across R2:
Vc1 = -46.6
Vvr1 + Vr2 = -(7.57 + 41.1) = -48.68. So we're close enough.

Last thing to check for voltages would be to make sure the offset voltage is getting all the way *back* to the transformer.
This one would much more difficult to determine if you measured it at the transformer but you might be able to determine this on the 6.2VAC input. When measuring the AC voltage directly between pins 1 and 2 - you should get about 6.2VAC (you measured 6.7VAC as you mention above so we're good there).

BUT when you measure the AC voltage from ground to these two pins - the AC voltage should be offset by -41.1V (VCO).
With one meter to ground - measure the AC voltage at pin 1 and then from ground to pin 2 of the connector.
Downside to this is some meters will only measure the AC content resulting in 6.2V (6.7V).
Some meters will display not only the AC content but the offset as well.
If your meter only measures the AC content of 6.2V then try measuring in DC setting. Yes - this will fluctuate but will center at the VCO voltage.

Yep, you're fine with those readings.
Well... that says all voltage to and on that board are good.

Next step says borrow a board if you can and try it out.

Different display.
His was MA-644, yours is MA-999

https://www.pinwiki.com/wiki/index.php?title=Gottlieb_System_80#System_80B_2

I would suspect your U4 and U5 were the replacements to the other board's Z1 and Z2.