we can argue all day long about broadcast mechanisms but if the origin is the original origin then definitely you want to broadcast them
group messages, maybe you want to only send them to group relays, but kind 0, 1, 10002s and all those other fun public things should be sent everywhere from the first second, second round, shouldn't be necessary if first round worked
when i say broadcast in practical terms that means epidemic transmission, meaning that you send them to several nodes, and say "plz share" and they send them to several nodes until everyone sez "stop sned plz"