How can I electronically generate a signal similar to the one produced by striking a piezo?
This is not a complete answer to your original question, but I hope it'll help you make some progress.

I've done a similar project on the PS4 version of this peripheral. In that process, I've tried to reverse-engineer the piezo input-stage. You'll be able to find a circuit diagram as well as pictures of that circuit at How does this piezo input circuit work?. The answers and comments also explain the function of the various components, but do note that my original measurements of the capacitors were wrong, so the time constants in some of the responses are off.

I also captured a few waveforms of strikes on the trigger pads. You can have a look at them at https://gist.github.com/rafl/c1f08aadad87f96bc741237a8c46b30d. CH1 is the voltage of the piezo relative to the controller's ground as measured at the connectors you mention, and CH2 is the voltage at the output of the piezo input-stage, which is accessible with test points. The file names indicate the time base and sometimes some details about the hit. You can use sigrok to visualise the data.

For the basic four pads, I've been able to reliably trigger the game with square pulses of 1ms width, and and a hold-off period of 40ms between pulses to avoid accidental double-triggering. With the input being AC-coupled, I guess the amplitude of the pulses doesn't really matter, but I've used 3.3V as that turned out to be most convenient for the micro-controller I was using. The actual logic signal level of the controller is 2.8V.

For the 3 cymbal trigger inputs, I've been able to reliably trigger the game with square pulses of 25ms width, and a hold-off period of 40ms between pulses. I used 3.3V again. These inputs differ in that they have no input-stage comparable to the one discussed in the other StackExchange question mentioned earlier. They might contain that circuitry within the cymbals, or be constructed differently. I don't have a set, so I can't check.

For the kick trigger, I've been able to reliably trigger the game with square pulses of 60ms in width, and with a 50ms hold-off period. Same signal level. There's no AC-coupled input circuitry either. I'm not sure why this input needs pulses so much wider than all the others with my current approach. I haven't checked how quickly one would be able to re-trigger the original kick drum pedal in the game. I believe the original pedals use a reed switch, so perhaps that behaviour is intentional to account for some inherent bounce from that. I'm also not sure if perhaps the maximum frequency of repeated kick drum hits in the game might be lower than the maximum for some other pads so perhaps this doesn't matter.

I've also been triggering the other "standard" buttons in much the same way as the real drum triggers described above. 1ms pulses work well again, and it appears like latency from those inputs all the way to the game is lower than when using the other trigger inputs. If you don't care about cymbals, these might be a better way to go than the piezo inputs.

I also think that reliably re-triggering at 10Hz is probably good enough to enjoy the game. You can see what drumming at 20Hz looks like at https://www.youtube.com/watch?v=Q9FrW-Wr-ds, and I don't think there's a lot of passages anywhere near to half that speed on a single pad in the game.