I was wondering how long a main control board should be expected to last on a Bottom Freezer Refrigerator? The model number is LFC23760ST

Probably a bad fuse on the main board. Fuse is not replaceable so either you have do a workaround (search LG refrigerator fuse repair for several fix videos and how-to's) or you have to replace the whole control board (part EBR41531302 I believe, but double check with your the part vendor before ordering).

Check the fuse by pulling the refrigerator out, unplugging, taking off back control board panel, and checking the fuse for continuity.

Compressors cool the fridge, the main controller does all the electrical commands, i'd check if the compressor is cutting in if it does then there is a gas or fan problem.

No need to buy a new main control board !
The problem is mainly from the fuse that soldered in the main control board, go to buy a "125V 15A or 250V 15A" fuse ($5.00 a pack of 5 pcs), cut the burnt fuse from the board, re-soldered the new fuse into the board, the refrigerator work again. you save $80 for the labour cost of the technician and $180 for the cost of a new main board.

If you have no lights or hear the motor running, I would check the house circuit breaker first and measure the voltage at the outlet. If that is within specs then the one thing that could cause your problem in the refrigerator is the power control board P/N: 6871JB2047A

slow

the man board is the problem.
replace it with new one..

LG LFC23760ST
I recorded a 2 part movie clip on how I repaired the blown fuses on my LG refrig's circuit board.

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You have a blown fuse on the main board. It can be replaced with a new 20 amp glass fuse using a soldering iron. Remove the cover and check the fuse if it is burned that is your problem

You have a blown fuse on the line incoming voltage circut. Remove the main board cover on the back and look for a glass metal ends 20 amp fuse. You can remove the fuse carefully with a soldering iron and solder a new one back in and you will be running again!

This is the main control board failing. My symptoms were identical (intermittent display and clicking), and eventually the clicking didn't go away and the freezer got hot. I replaced the main board yesterday and all is well (display works and no clicking). I was able to find the correct part number using the GE website (entered refrigerator model PD22S1S and then selected PDS22S1SARSS from the drop-down list). My part number was WR55X10656. The GE price was shown as $157.50, but I bought mine from ApplianceZone.com for $101.68. Hope this helps.