Why webmail runs out
Free providers make it slow to open lots of accounts, and new accounts often get extra checks the first time they're used. Making twenty addresses the night before a drop is a miserable way to spend an evening.
What resellers use instead
A domain of your own, so every address you need is yours and lands in one mailbox. Email forwarding, so new addresses still end up somewhere you check. And one inbox view for confirmations and codes from all of them, 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 ready before the drop, confirmations and codes in one inbox view, and every order showing up on your tracker without any copying.