So, this sounds like a job that multi-Opamp packages were invented for.
Take any modern opamp (please, stay away from 1970's opamps...). I've clicked together a TI.com search that gives you candidates. Since you seem to be planning on ordering a lot of them if your design works out, don't hesitate to order free samples from TI.
So, I'm not going into detail on the individual OpAmp circuits, they're easy to google everywhere, but to map your signal flow to OpAmp configurations:
- Op Amp buffer
- envelope detector
- slope detector
- rising edge detector
Google the terms in italics
simply a voltage follower. Might be superfluous, if you properly design the next stage. You might want to AC couple (capacitor in series with) the Piezo into a high-ohmic Voltage divider, so to bring your signal around a virtual ground (most often used: Vcc/2).
An opamp in what is commonly called a precision rectifier, followed by a low-pass filter. Go for something that let's you easily define a ground level without loading that ground, so that, again, things are centered around virtual ground (Vcc/2).
High-Pass filter. You can implement that as active filter (e.g., Sallen Key), but a simple RC high pass might totally do here.
You'll notice that this is a high-pass after a low pass. Simply substitute both by a band-pass if easier.
Rising Edge Detector
Rising Edge in Envelope -> positive voltage in derivate. The Slope Detector should act roughly like the derivate of your envelope signal, so just make a sign decision against a virtual ground voltage. That'd be a comparator.
You might want to not only compare to virtual ground, but also add some hysteresis. Go for a Schmitt trigger.
That's one quad opamp, two diodes, three or four capacitors, and something between 7 and 10 resistors.
You could have the same functionality with a single capacitor + two resistors for AC coupling the Piezo (and low-passing it "by accident"), which has plenty signal amplitude (wouldn't have thought it'd be that much as in your plots), directly into the ADC of a cheap microcontroller (I don't personally like the ATTinies overly much, but this might be a good application for one).
You could also just see how far you can get by simply using a cap to couple the Piezo signal into the resistor-biased base of a NPN transistor in common emitter configuration, filter the collector voltage with a passive bandpass and then generate your output pulse with another NPN. That'd probably do, too, but I don't know how resilient you can get that without a lot of tweaking and testing.
In the end, I'm also personally not much of a Through-Hole person. I see the ease in soldering in wired resistors compared to 0603 SMD resistors, certainly, but maybe you'd want, for cost reasons, just use a SMD IC and the discretes in through-hole.
because I need to build a lot of them
if "a lot" is greater than, let's say, 100, consider building a prototype, then convert that to SMD completely, and order the thing being manufactured.