Exclamation Fridge Not working|Tested components with Multimeter |No luck

Well, if it's cooling then there is a pretty good chance that the compressor is running and it's relay is working correctly. You wouldn't need a multi meter to figure that out, but whatever. You said "all compenents are fine and give appropriate readings". While there you go, end of story. That is unless you were lying about all components being fine. Can we trust you and your multi-meter using abilities?

You also said something about no ice buildup. Does this mean that you actually took stuff apart inside of the freezer compartment and found the evaporator coil and inspected it?

If so, then you can rule out any issue with your defrost circuit and/or evaporator fan.

Pretty much the only thing left would be refrigerant charge and/or a restriction in the refrigerant circuit.

If this is the case, time for a new fridge.

Leaving the door open could have caused the evaporator coil to frost up and block all air flow. If you did that for a long time, the frost buildup can cause ice to form. Ice isn't easily melted in an ordinary defrost cycle and may require getting access to the coil and using a hair dryer (I've also used propane torch) to manually defrost it or else putting the unit out of service with the doors open for a couple of days.

I'm hoping the click sound you heard was the defrost timer and not the overload Klixon. If it's clicking repeatedly all the time, you should unplug it and have a pro check it out. It may be fixable since testing with a meter isn't very definitive when it comes to a compressor that won't start. The best test is a substitute start pack which a pro will have or know how to simulate.