piezo element vs accelerometer
Some accelerometers use piezo elements, though I think nowadays they tend to use MEMs.

The main reason accelerometers are so ubiquitous is because a great number are made to serve the smartphone market, and smartphones are ubiquitous.

The main functional difference is that MEMs can be sensitive to DC while piezos are only sensitive to AC. Not good for detecting smartphone tilt, but is besides the point when detecting an impact.

To use piezos for your application, I believe you have to prestrain them against an inertial mass. After all, it's their deformation that produces a signal. That could be tricky.