You or I can post a free add on an amateur archeology FB group looking for takers to keep stuff out of the dumpster. The government doesn't roll that way, especially when it comes to property that was evidence in a high profile case.
Of course the nice stuff will find homes in museums but nobody has a collection that's all nice stuff. The pristine pot will wind up in a museum. The pottery shards that no museum can justify the space for but some small time private collector would want will probably mostly be trashed. If we're lucky some intern will be tasked with taking a ton of ultra hd photos on the way to the dumpster so they're not completely lost forever.
The government simply doesn't have good mechanisms for getting rid of stuff like this (whereas private individuals will happily send a flat rate box to whoever paypals them $25 if it means sending part of grandpa's collection to a good home). Seeing as this is a high profile collection they may expend the effort but I wouldn't hold my breath waiting for it to show up on govplanet.