> This leads me to an idea which, I don’t believe is typically done, but which I don’t know why not: Why wouldn’t a company just, list a price that they are “going to sell it at”, and then “sell” a large proportion of them at that price to an entity which isn’t publicly connected to the company, and then that organization sells the items via auction (and returns a large proportion of the profits back to the company).

That's often how a lot of today's "scalping" is done - AIUI a lot of "secondary market" tickets on StubHub etc. are sold by an entity that's owned by TicketHub or whoever, so that the artists can sell their tickets at the price the market will bear but put a low face price on them for the sake of their reputation.