You explained that the design...
is mostly copied from the datasheet example
Personally, when starting to use an unfamiliar chip, I would often begin by completely copying a simple datasheet example circuit before any experiments or customisations :-) That allows us to confirm that our specific chip is working OK, and gives us a known-good "starting point" we can go back to, if things stop working during our experiments.
The biggest issue I see is where your design is different from the datasheet examples - your use of pin 7 (ST-BY) is not in accordance with the STA540 datasheet I found here:
Check Figure 19 on page 14 - by pulling pin 7 to Gnd (0V - left side of the x-axis of the graph), you will have maximum attenuation so no sound! That fits exactly with your problem that...
there is nothing on the speaker output.
Also note that you must follow the specific requirements for driving pin 7 (ST-BY) as explained in sections 6.4 and 6.5 in that datasheet:
Some precautions need to be taken when designing the driving circuit
for pin 7, ST-BY. For instance, the pin cannot be directly driven by a
voltage source having a current capability higher than 5 mA. In
practical cases a series resistance must be inserted, giving it the
double purpose of limiting the current at pin 7 and to smooth down the
standby on/off transitions. And, when done in combination with a
capacitor, prevents output pop.
A capacitor of at least 100 nF from pin 7 to S-GND, with no resistance
in between, is necessary to ensure correct turn-on.
See the example circuits in the datasheet for suitable RC values if you want to be able to control ST-BY externally, or use a 100nF capacitor only (no resistor) between pin 7 and S-Gnd as explained in the above quote from the datasheet, to at least get the chip out of standby mode. That would be my next step in troubleshooting (although there could be other problems e.g. regarding the wiring between P-Gnd and S-Gnd which we can't tell from just a schematic).
I hope the current circuit hasn't damaged your chip... Good luck!
Added: Sparkfun produce a kit using the STA540 and their schematic is available here, for you to see a known-working example: