Can you advise where you got the wheel from? What size are your original rims?Agreed, the pump is useless in most cases. If you see a tire getting soft you can just go to a gas station or convenience store for air. Who needs the pump? My tire place adds air for free, and also does plug & patch for free.
If it's a catastrophic failure like your wife's tire was, you have no choice but to use a spare. If no spare you're sunk.
Your dealer will promise to order a spare tire kit for you, but it will never come. I've been waiting since 2018. If it ever comes I'll offer it to you first! I made my own kit by saving the best of my old tires, buying a wheel with the same bolt pattern ($100). I already had a cross wrench and a scissors jack. I put the air pump in there, too (why not?) and there's still room under there for some basic tools and misc. stuff.
The reason there's no spare is to reduce weight in order to improve fuel economy. Just think of all the gas you'll save! And how much fuel AAA will use sending a hook to rescue you. Kinda like the auto stop/start system. It saves gas in the short run, but wastes it in the long run. I digress. I'm glad your wife made it home safe.