The 2nd Generation Airport Express has the combination optical and 3mm mini analog jack for attaching speakers or AV receivers. NOTE: The 2nd Gen Express implements Airplay1 and seems to uncomfortable with Airplay2 (at least on the latest MAC OS versions.) It appears that something about Airplay2 is not allowing the older Airplay 1 systems to be compatible. The device connects with some sources on the MBP OS with >Version 11 but others remain silent or sometimes connect and play only for a few seconds.
My solution to the compatibility problem is to use Rogue Amoeba’s Airfoil product (which also allows output to two or more Airplay devices at a time.)
For you Hardware Question: I have two of these… one uses optical output and the other the analog output. Apple shipped the 2nd Generation Airport Express with a special cable that has an optical probe that fits inside the 3mm jack on one end, and the standard TOSLINK connector on the other end.
When playing audio through the unit, you can see the red LED light inside the jack when there is no cable connected.
I am not clear which problem you have. If you see the light, then the optical output is working. However, a regular TOSLINK optical cable will not fit and not work event if you taped the connector over the hole. You need the special cable connector from Apple (probably not available anymore).
If you don’t see the red light, then the optical output is probably broken. However, this would be an uncommon failure mode and having taken an Apple Airport Express apart, that would be difficult to fix. You would likely have to convert back to the analog 3mm stereo jack to use the device.