I'm guessing that is the remote sender board, it appears it isn't sending any RF signal or the receiver isn't picking it up. With only two units it can be difficult to tell which end is responsible, when you say the receiving unit sounds the melodies, how are you triggering them?
The Tx PCB looks fine to me. I normally would suspect a bad push button. But as you said the LED lights up OK, then it should be fine too.
The inductance are printed on the PCB and I couldn't find datasheet for MC908T and the SAW. Did you do a rough check on Q1 and Q2 BJT to make sure that they are not shorted?
Your PCB is attached here..
Another similar sender I found which is similar uses 433MHz frequency..
I had a second transmitter and it worked fine with the receiving unit.
I suspect, but have no evidence, is that the first transmitter failed due to the high humidity environment of living 1/4 mile from a lake fed by saltwater.
I have a near constant breeze.
I may try to find a way to make the transmitter water proof or at least water resistant.
I live on the coast with constant seawater nearby and in an area of very high rainfall, I have never had problems with corrosion.
If the LED on the sender is coming on when you press the button, the fault almost certainly lies with the transmitter oscillator. That would be Q2 or a component connected to it. From tracing the tracks it appears the signal that turns the LED on is the same one that enables the oscillator (ASK mode) so if one works and not the other it implies the oscillator is at fault. You need an audible receiver, spectrum analyzer or a wide bandwidth oscilloscope to check the operation conclusively.