Wouldn't that be the most difficult and expensive way to simply monitor for life or technosignatures? Compare that to simply (relatively speaking) building massive space based optical and radio observatories that can observe every visible star. The light reaching you has already made the trip, you just have to collect it. If you were murderous, you could launch an attack from your home system rather than waiting for the probes to spread from system to system.

Granted there could be some technical limitations that prevent you from observing certain systems but that pool will be much smaller than an entire galaxy of stars.

Also, don't forget that a Von Neumann probe is at a disadvantage when it arrives in a system since you want to send the minimum number of probes to each star and then have them replicate on arrival. If your probe accidentally shows up in a system that contains an advanced civilization capable of detecting and capturing it then you've given away the fact of your existence and your intent along with all the information that can be gained by studying the probe.