If I play Promising the Future and choose to play Dazzling Aurora, my opponent chooses to play Brynhildr Thundersong, what happens?

Yes — your Dazzling Aurora still gets played and enters as a Gear.

Because Promising Future says "starting with the next player," your opponent plays Brynhir first, then you play Aurora. Both go on the chain as pending items before Brynhir's "When you play me" trigger can do anything. Brynhir finalizes first and her "opponents can't play cards" effect goes on the chain — but it only stops new card plays. Aurora is already pending, so it finalizes normally and attaches as a Gear (RiftJudge FAQ #229, #11870).

One practical consequence: Brynhir's restriction stays active for the rest of the turn, so when Aurora's end-of-turn trigger later reveals a unit, that unit can't be played and stays banished.