SBD is bidirectional unicast (and mandatorily so; as you say, SBD can't transmit to devices before they initiate a mailbox exchange, with the exception of a "message waiting" indicator.

Iridium Burst seems to be multicast-capable, and optionally bidirectional or unidirectional, so "spicier SBD" makes sense!

But their paging service was definitely unidirectional and unicast. Here's an old talk on decoding the signal, if you're curious: https://media.ccc.de/v/31c3_-_6236_-_en_-_saal_1_-_201412281...