EDIT/NOTE: before trying what I said to do in this post jump down to my later posts and try that stuff first. WHEN TROUBLESHOOTING, ALWAYS START WITH THE EASIEST STUFF! See my very last post about the CHNT 32X fuse holder and make sure that fuse is good and able to pass current (continuity test with your multi-meter). Well, now we can deduce that your electronic control system appears to be functioning like it should - when the status of all key parameters (cooling system on, not too hot, not at cut-off pressure), the relay is closing which
should allow passage of 120 VAC to the compressor motor.
And the AC side circuit breaker appears to be good.
And we know that the system is receiving AC voltage from the wall socket, since that is what is powering the AC/DC Converter.
BUT the AC voltage is being interrupted somewhere between its source and the compressor motor. One interruption point is the main switch, but you have already replaced that.
Do you see the black plastic box on top of the compressor motor, above the left blue arrow in the photo I provided above?
Yours likely has one. Within it reside a couple of big capacitors. They 'soften' the impact of compressor motor start-up on your house circuit. Motors draw the highest amperage when they are just starting up/moving very slowly, and this can trip a home's normal 15 or 20 amp circuit breaker. The capacitors prevent this from occurring (at least as often as it would without them).
If your compressor has such a little black box, the main 110 VAC feed from your control system goes into that box, to the capacitors, and then directly to the compressor motor (internal wiring which you cannot see.
If you can find the thick cable which goes into that black box, you should trace it back to where it is connected into the control system. If convenient you can disconnect it there, and wire-nut a spare wall plug (from an old computer or appliance? I save that stuff, but I know that not everybody does), and plug it directly into a wall outlet. This is treating your compressor motor just like it is a toaster - plug it directly into the wall and it will either work (start to run) or not. If it runs, we have confirmed that the compressor motor is good and we continue to troubleshoot the control system backwards from that point. If it doesn't run, then the compressor motor is the issue.
Perhaps lots of stuff and maybe getting out of your comfort zone a bit (?), but in the end you will know your equipment much better!