A failed super capacitor does not normally stop a dashcam from booting, but maybe it is possible.
Can you measure the voltage across the pins of both super capacitors (2 measurements), when the camera has power and also when the camera is not powered (4 measurements total)?
If the capacitors are dead, you could cut one of the legs, and see if the camera will then boot. The super capacitors are not needed for the camera to boot. (Cut the legs near the capacitor, not near the PCB, then you can solder a new capacitor onto the old legs, less chance of damaging the PCB than if you desolder the capacitors. Also, it is then possible to easily reconnect it with a little solder.)
When powered, the super capacitors should both measure about 2.6V, when the power is removed, they should both drop to about 2.4V and then with the dashcam powered down should slowly drop to 0V over a few days.