In principle that is a good idea and nice digital version of analog pulse shaping. However, the pulses have a very distinctive shape (not visible in the blog post picture, but here in figure 7/9: https://www.mdpi.com/1424-8220/19/19/4264).
My programs use the trigger level just to select which waveform snippets are recorded. The peaks themselves are then identified by their steep slopes which shows up nicely in the derivatives: https://github.com/ozel/DIY_particle_detector/blob/90c180207...
That is a typical kind of shape for a particle detector, though the time constants are very long which is a good thing because it's the result of the preamp doing some shaping to remove high frequency noise, which helps make the numerical integration less essential. However, I think you will find a trap filter will allow you to push to lower trigger thresholds more reliably. Calculating the slope from one sample to the next is equivalent to the output of a trap filter with peak time of 1 and gap time of 1. Once you enter the regime where the signal amplitude at each recorded sample is comparable to the magnitude of noise, especially for slow signals and fast background, the derivitive without any integration will start going crazy. What should matter most is recovering the area under the signal peak independent of noise on top of it, which will ultimately average to 0. An alternative method is to just integrate the entire signal over a window and then do baseline subtraction, but this still requires a robust trigger signal to indicate when to start integrating, which brings things back to the trap filter.
The pulse detection works really quite satisfactory with the current method and for beta-decay electrons and alphas. I also prefer that the derivative is easy to understand for students since this is still mainly an educational project. For detecting lower energy particles like fluorescence x-rays, your proposal would be certainly something to investigate. However, for improving the SNR I would in general first redesign the amplifier. It's really optimized for super low part count and costs at the moment.
That said, I will definitely keep you suggestion in mind. Thanks!