Aspire 7750-6423 backlight is out. Tried keyboard tricks and BIOS load---negative result.

Did you also replace the inverter? If not, that could be the problem.
However, as the problem is intermittent, it is most likely a poor connection or broken cable. If a visual check doesn't reveal the fault, you could try a continuity test. Another test is to partially open and close the lid to check for on/off or flicker which would also indicate poor connection, broken cable or fractured ribbon cable.
Personally, I wouldn't suspect the motherboard until all the above have been thoroughly checked and proved to be good.
 

Unfortunately, I can find no information whatsoever on your model.

If you are able to get the panel manufacturer and part number from the screen, you should be able to find the data sheet. From that point it is reasonably straight forward to test all connector pins.

If the backlight is the only issue you should be able to confirm data is being displayed by shining a torch on the screen. If your panel does not employ an inverter then, on the face of it, it does suggest power is not getting through and the data sheet will indicate the power pins to test. If those pins appear ok, it does suggest an internal problem which would probably require panel disassembly. However, as you have a replacement screen and both exhibit the same problem I would still suspect a cable problem. I say that simply because the panel did come back to life and, in my experience, this is not at all common where component failure is the cause.

Is it possible for you to bench test your original panel? Can you find the backlight connections/requirements (from the data sheet) and apply the correct voltage/current and see if it works?

Clearly, I have no way of knowing your ability level so I must stress that any action you take is entirely at your own risk. I am not recommending you do this - that is a decision for you. If your panel does have an inverter (and even if it doesn't), you may expose yourself to high voltages that could kill!!!

Afterthought:
I assume your model will have a function key to switch video output. It may be worthwhile taking a look at that too (although I would put that low on the list).