First , you are correct in the way you understand your probe. +5 / +12 , TTL , CMOS , you have it all correct. Stay on pulse as this would show you if the pin is going up and down really fast, both LEDs would light up at once. You are also correct that if all switches on row 4 work (all 8 ) it means that the entire column has to be working. Which narrows it down to the rows now.

Before we go on , you did verify you do have a stable +5v and +12v , yes ? (We'll assume it's good).

If we look at the schematic , we can see that the columns are outputs from the MPU and that the rows are actually inputs. If you mean to probe the columns , you would start in TTL , hooked to 5v , and probe pin 8 of U14. The columns are constantly driven one after the other over and over , so on this pin you should see a pulse on your logic probe. This pulse should be found again at the output, pin 9. This chip will buffer the signal from the MPU and also provide more current so that it can drive U20 that is right after it. U20 simply contains transistors and an inverter (the circle at the tip of the triangle means the outpout is inverted). Those transistors can provide even more current and they feed all the switches in the playfield. The input and the output of each line would also pulse , the output being reversed from the input. (You can't notice that on your logic probe , you don't really need to , but you want to see a pulse there too). So in this case we started by probing closest to the MPU going towards the switches.

To test the rows , the process starts at the other side. If you are testing Row 1 , column 1 and that switch is closed on your tester, you should :
Connect your logic probe to 12v, CMOS , Pulse. Now probe the banded side of D3 , which is the connection to the switch directly. This should be LOW , since the switch is connecting that pin to ground. Now probe pin 11 of U18. It should be low when switch #11 is closed. Releasing the switch should drive the signal back to high. (If this state doesn't change , something is wrong). Just to make sure you are using the probe correctly , you can try a known working switch like Row 4 column 1 , probe the banded side of D6 with #41 closed and it should fall low.Then probe U18 pin 5 and switch on and off , the state should change. When you've verified that the + side of the opamps has its state changing , probe the output (pin 13 of U18 for switch #11 and pin 2 for switch #41). You can leave your probe plugged to +12v and CMOS as the output (5v) should be enough to drive the CMOS circuit and give you a HIGH output, a low output will always work whether it's set to TTL or CMOS. If you can't measure the output , move your probe to 5v TTL , but it should still work anyways. Now , unless U13 is pulling down the output of the opamps to a point where your new U18 isn't working at all (doesn't seem the case since row 4 works), the outputs of U18 should be toggling high/low when you close switches. Go ahead and measure pin 11 of U13 after you've tested pin 13 of U18 , they should be identical since they are directly connected together and to nothing else. If you verify that pin 11 of U13 changes state (5v , TTL) then next is to check the output of U13 at pin 9. Again there is a circle at the output ,meaning the output will be reversed to what you read on the input. The state of this pin should change when you press the switch.

Beyond that , pin 11 of U13 is connected to the data bus , and the data bus is not affected otherwise the game wouldn't even boot properly since all data is on that bus and components are disabled when another uses the bus. having all columns work on row 4 is another indicator that the data bus is correct.

Of course all this trouble shooting assumes that all connections have continuity across the different chips and that there isn't any corrosion damage on the board. (a problem that's very frequent and affects the switch matrix). Can you post close up pictures of that board ? Some might see things you have missed.

To quickly summarize , if you probe 5v TTL pin 11 of u13 and the state changes when you close the switch (#11) , but pin 9 of u13 doesn't when you are closing the switch , then U13 isn't doing its job correctly. Otherwise , you have to start from one side and follow the signal through the chips and to the next one, on and on..