OK -- figured it out, it's doing what it's told to do but not as expected. As you said - it it going through the subsequent tests. Bad Z14 was pulling one or more of U4's switch matrix inputs high so it sees the start switch as being pushed (Start = Proceed to next test).
First test:
"4", "5", "6", "1" -- main ICs check out fine.
Test fixture then checks the switch matrix. If Row 4, Column 7 shorted (Start switch pushed) -- it proceeds to test 2.
Test 2 displays "-2-" (dash then 2 then dash) and then steps through lamp control (LD) values ("1", "2", "4", "8"). Normally this would go 1,2,4,8 twelve times before restarting test 2 -- once per lamp strobe (DS) value. But after first set of lamp control values, it checks the start button to see if it should proceed to next test. If you had this board connected to VFDs then you would have seen the displays show test 2 and count off control and strobe values for displays.
At end of first LD set -- QS80 thinks start button is pushed so it goes to test 3 ("-3-") where it counts off test 3 steps.
Same to test 4, 5, 6, etc.
Tester was just getting into to the switch matrix test "-5-" when you shut it off. It showed "C77" which said Row 7. column 7 are shorted. Looked like it was getting to Row 6, 7 ... but that is when it got shut off. I'll bet if you put the bad 7400 back in there -- all column 7 switches would show up as closed. Maybe columns 4, 5 and 6 as well but wasn't enough of the video to show that.
I don't remember where the machine's 'test' button is on the matrix but I think he's also column 7. This controls how fast it steps through the tests. I'll bet it kept reading that button as closed and was going up and down with the speed of the testing as well.
Not having the VFD's hooked up makes that board a bit difficult to decypher but info is there. Next gen board adds 2nd display - bit easier but still easier if you have VFDs hooked up. I would love to put a simple USB port on this board to spit out real information but the amount of memory space available is way too small. Not enough space to fit the tests plus any text.