> I assume that this means you're sending out rejections that include a mention of "we've hired someone else for this role".

No, this isn't possible unless you delay rejections letters until you hire someone.

We send letters as soon as the decision is made not to continue with that candidate.

Honestly it would be cruel to string them along any longer.

From applying places recently I'd much rather get these fast. One company sent me them, the rest either reached out or I never heard from them.