Ok, there's a few things we should look at.
The first, as you've pointed out, is U33.6 being close to 0V. Because this pin was changing slightly before, the pulldown for that node might be open (R62.5). The easiest way to check is to use a 470 ohm resistor and hold it to pin 6 and Vee (pin 8 ) at the same time while watching the voltage on pin 6. If you get a normal ECL logic level, R62.5 is open. You can either replace resistor network R62, or just solder in a stand-alone 470R resistor underneath. I would probably do the latter since it's easier and you can move on to other areas.
If pin 6 stays near 0V, it's likely the output on U32.5 is toast.
What is the voltage at U33.5?
On U37, three of the four outputs, pins 2, 3, and 15 are unusually low. The fourth, pin 14, is inconclusive because it is wire-OR'd with two other gate outputs. I am highly suspicious of U37. The reference voltage generator in the chip is shared between the gates and it might be dead.
A possible way to check is to make the unused output, pin 9, live by connecting it to Vee through a 470R resistor. See if you get a valid ECL logic level on pin 9. If not, it may be time to pull U37.
If you remove U37, I'd recommend 1) put a socket in its place, and 2) check the voltages on all the pins while it's out because the isolated readings may provide clues to other problems.
I'm also noting that U37.2 is wire-OR'd with U33.9, but other sections of U33 appear to be working, which goes against the dead U37 reference generator theory. I'm not sure what to think of that yet, but it may also indicate an issue with that gate on U33. Measuring the pins on U37 with U37 out can tell us more.
And as with placing a 470R on U37.9, it may also be useful to make U33.15 live and get a voltage reading on that too. (U37 and U33 pins 9 and 15 should be complements of each other.)
Another approach when there are many logic gates hanging off a single node, like in this circuit, is to use a logic pulser and current tracer combo, such as the HP 546A (adapted for ECL) and HP 547A. These allow you to determine where the current is going. They can be very useful with ECL with its wire-OR'd architecture, and can often lead directly to the culprit. If you have these, let me know and we can make use of them, but they are not common. They can be had on ebay, but they command a high premium and not worth it at the going rate, in my opinion.