"I have watched the troubleshooting video. I do not get any noise touching a probe to the center conductor of the SO-23 ..."
You need to start the test at the LM386 audio stage. Farhan's video using a 6ft length of wire is good because the wire will pick up (low frequency) hum and wide band rf signals so you can move the test point back from the audio stage towards the antenna. The noise from the speaker will change but you should get some sort of noise from the speaker at each stage. Where it drops out identifies the faulty stage.
If touching the input to the LM386 does not give noise that is where you need to start looking for a fault.
If you have a signal generator that is best but you can get a lot of help just from a 6ft length of wire.
Good luck. Brian