Okay, I hoped to see an RC reset generator for the MCU - these things can fail and prevent the chip from booting properly. But it seems there is none here.
C3, C4 are in parallel between 5V and GND, aren't they? Boring stuff.
C8, C9 seem to be load capacitors for that white tube crystal/resonator, whatever it is. Speaking of which, if you have a scope, you know what to do. In fact, even if you don't, at least check what voltages are on each end.
If C2 is also between 5V and GND, don't bother with it. Its value isn't very critical. It does no harm unless it's shorted. Temporarily solder any cap of at least 47uF/10V in parallel with it and try again.
I think C2 is 47uF/16V, by the way. Looking at similar caps on the main board may help in identification.
Inputs to the HC05 are pulled up by 10k resistors to 5V. Which means, maybe the big chip drives them high or maybe it doesn't even drive them at all (like because its firmware doesn't boot and the pins are high impedance by default). See if connecting a 2k2 resistor from HC05 input to ground pulls it down to 1V and flips the inverter or if the input still stays hard at 5V.
Check if those HC05 inputs aren't permanently shorted to 5V. Check if there is continuity from them to some pin on the MCU. But really, I doubt that anything is wrong here.