Why webmail runs out
Free providers rate-limit signups per phone number and per device, and they start asking for verification after a few accounts. Making twenty addresses the night before a drop is slow, and half of them get locked the moment they're used from a new location.
What resellers use instead
A catch-all domain: one domain you own where every address receives mail, so you can invent addresses on the spot and still get the confirmation. Apple's Hide My Email, which creates real, deliverable addresses that forward to your inbox. And a proper inbox that can read confirmations and verification codes from all of them in one place, because twenty logins is not a plan.
Having enough ready
The trick is preparation, not speed. Generate what you need before the drop, keep the confirmation emails somewhere you'll actually see them, and make sure whatever account you're using for the order can receive its own verification code without you logging into another mailbox.
For Minted members the email side is handled: addresses generated ahead of time, confirmations and codes readable from one inbox view, and the same inbox feeding your order tracker so every order shows up without any copying.