Why isn't this single pushbutton on/off circuit working?
Sometimes, after you've carefully checked everything, it turns out to be someone else's fault. In this case I think the original design is bad. I've tried it in Circuitlab (should be here) and I've tried it in Falstad:

1 5.0E-6 85.40587625261516 60 5.0 50
r 128 112 128 208 0 470000.0
r 128 208 128 288 0 470000.0
r 400 96 400 192 0 1000000.0
r 400 192 400 272 0 100000.0
r 272 208 208 208 0 10000.0
r 272 224 272 288 0 10000.0
r 272 288 272 368 0 470000.0
r 272 288 336 288 0 1000.0
t 336 288 400 288 0 1 0.5854708958314027 0.587099911972777 100.0
t 208 208 176 208 0 1 0.587957264003006 0.588497761400722 100.0
w 272 224 272 208 0
w 400 304 400 368 0
c 128 288 128 336 0 1.0E-6 0.02827590396558895
w 176 192 128 208 0
w 176 224 176 368 0
w 128 336 128 368 0
t 304 192 272 192 0 -1 -0.5251112318758011 -0.5895458111379241 100.0
w 400 192 304 192 0
w 272 176 272 112 0
w 128 96 128 112 0
w 272 96 272 112 0
w 400 96 272 96 0
w 272 96 128 96 0
w 128 368 176 368 0
w 176 368 272 368 0
w 272 368 400 368 0
s 208 288 240 288 0 1 true
w 272 288 240 288 0
w 128 288 208 288 0
v 528 160 528 96 0 0 40.0 9.0 0.0 0.0 0.5
w 528 96 400 96 0
w 528 160 528 368 0
w 528 368 400 368 0
w 400 272 464 272 0
o 33 64 0 35 0.009765625 9.765625E-5 0 -1
o 28 64 0 35 0.3125 9.765625E-5 1 -1


In both cases it doesn't respond to button presses (this is rather hard to simulate in circuitlab)

Without any more "theory of operation" info (no I'm not going to bother watching the video), I would say there is a design problem. A more conventional "bistable multivibrator" might work better.