Piezos can be very fast, however the sound inside a tin foil also has a finite speed and will thus add to errors. So it would kind of need two piezos on both sides.
There will be reflections from the projectile, and this can be rather tricky. Working in reflection effectively double the distance. Thus focusing and adjustment get more difficult. A retro-reflector helps a little - otherwise the angle of the mirror is important too.
One could modify the shown circuit by adding a second detector, thus making targeting less critical. One could likely still use a single IR LED. Even if the detectors a separated a little, the sensitive area can be overlapping and without a gap.
Even if using a comparator for triggering, one should consider also capturing the amplitude, as a partial hit could also cause a wrong timing.
In school we did an experiment with two thin strips of Al foil, that get separated and detected by interrupting a current flow. So it needs disposable materials, but that is cheap.