Thanks for all the responses! We went for the MP card, 50k miles! Apparently they can expedite it, but perhaps not all the agents know this. Our bizarre timeline:
Thurs, May 21: Instant approval, asked for expedited shipping. Agent 1 said it is not possible, we'd have it in 7-10 business days.

Weds, 26th: Boyfriend got an email that the card was in the mail.

Mon, June 1: Boyfriend called, the card should be here by now..? Agent 2 says it went in the mail Friday, and has no idea why Boyfriend got an email about shipping. Also that there's a little box to check to expedite the card, but Agent 1 hadn't checked it. Agent 2 was surprised and annoyed by this. Boyfriend added me as an authorized user, agent clicked the "express" box.

Tues, June 2: His card finally arrived. 8 business days after approval and one day too late to pay for the workshop that would've covered the $3k in one purchase!

Thurs, June 4: My card arrived in the mail, along with another card for him!

So, a cheaper card can be somewhat expedited, but I wouldn't rely on it!