No, installing C3 with incorrect polarity will not damage Q9 or any other component except possibly C3 itself.
Are the voltages on Q9 in line with those on the schematic?
If you don't have a scope to see if it is oscillating you could try this.
1. Get a working radio with AM band (550 to 1600 kHz).
2. Turn off all other electronic apparatus in the vicinity. Especially all switched mode PSUs.
3. On the working radio find a quiet frequency anywhere between 1100 and 1600 kHz. Note the frequency. Turn up its volume so that you can easily hear the background static.
4. Place the two radios as close together as possible. Turn on the faulty radio, switch it to AM band and turn its tuning knob around the frequency noted above minus 455 kHz. If you hear the static of the working radio decrease and increase as you tune the faulty radio then its oscillator is working. If so, check that minimum static occurs when the frequency difference on the two tuning dials is approximately 455 kHz.