No, I don't think that it's common for a new board to go bad but there are things that can kill a brand new board 5-seconds out of the bag. Stray electrical discharges or shorts can ruin any electrical circuit especially one that doesn't have robust safeguards built in. That's not a feature or specification you'll find listed but sometimes more expensive electronics will have better safeguards against accidental surges or static charges. It's possible to short a board without knowing it if you have power supplied to it while changing things around. Changing an extruder should not kill your board so unless something odd happened, I'm still thinking it's something simple that is being overlooked.

Ok, the motor is good so the issue is somewhere else.

Have you tested the continuity of the wires that connect to the board and then to the motor? I assume your motor wires are not long enough to reach the board and if not, you'll certainly have another set of wires to extend that length. If those are all good and none shorted (make sure you wiggle connectors as you test) then we can eliminate wires.

If the wires and the motor are ok, it starts to lead to the board.
1. Did you change anything in the firmware that could have disabled the connections on the board?
2. Are all of the stepper connectors dead or just one of them? If just one of them is dead can you move the wires from it and have that motor now work from a different connector?
3. Don't be insulted as I'm just diagnosing with you, but, when you've been making connections, was the power off to the board? Did you touch the board and at any time get a static discharge or shock? I'm trying to see if you board could have been damaged by stray electrical events.

I'm not an electrical engineer so I'm not sure what the stepper motors output. I'm not sure if it's continuous current or pulses. If you can figure that out with a Google search and you have a multimeter you should be able to set the printer to do a print that and then measure what's going on at each stepper connection. If you find that they all work except the one that seems dead, and you are sure nothing in the firmware was changed that could have disabled it, I'd say the board has a fault.

Does the board have a 'reset' button. Some do (mine does). Try that while it's powered on. Also, you can disconnect all power leads to the board and hit the reset button? I've found that it will open a circuit that will drain any residual electrons out of the circuits. Then, hook up the wires and see if the board works. I've reset a few things that way. I had a motherboard that would get a headache and not boot or turn on. If I disconnected the power connector and then hit the 'reset' button I would get an instantaneous blip of power that would make the power leds come on for an instant and then go out. That would drain the board. Then I could put the power connector back and it would boot. It was an intermittent thing and frustrating but it worked.

Again, I'm just trying help diagnose. You have to eliminate everything you can before you toss the board, jsut to make sure you aren't throwing out a good board by accident.

I'm not sure if I helped much here but maybe I gave you a few more things to try before you give up and order a board.

Finally, many of use RAMPS boards on an Arduino Mega. It's a robust board and not too expensive. You might be able to migrate to one. My firmware is Marlin and it runs well on it. I'm not sure if other firmwares work well with RAMPS but maybe someone else here will know that.