At the time of writing the original post, I was unable to retrieve all berry species of the identified flowering plant orders within a reasonable time. After all, I am not sure why this happened. However, it is important that the implicit EntityClass functionality works, so I tried again today and with susccess. So this is an addendum to the post above: the code to retrieve all common names available for berry plant species associated with the "flowering plant" orders is:

BerryCommonNames = 
 EntityValue[
   EntityClass[
    Flatten@EntityValue[order, 
      EntityProperty["TaxonomicSpecies", "SpeciesList"]], All], 
   "CommonName", "Dataset"] // DeleteMissing