They have sold out in the past, different years have shown different results I believe last year seemed to last the longest though that's just my opinion.
Your summary is correct you can either use your MB and have things charged to the room and then later visit the front desk to pay the charges off with the gift card. Or you can use the giftcard directly at a point of sale and bypass having to settle up at the front desk later. If you save the gift card info on your phone you can use it for mobile orders by manually entering the number and pin at checkout.
Also if you purchase a digital gift card from the
Disney store those can be added directly to the digital wallet on your phone and you can use the Disney gift card website to combine cards onto that up to the maximum of $1000. This gives you a digital bar code that you can have scanned at the register so that you don't have to carry around a physical card, or you can copy the card info off of this in order to place mobile orders.
Any method works and I used all three in a recent August trip to WDW, you don't actually have to stop at the front desk daily in order to settle the room charges. And even if you miss one before it has been processed they will process a charge back for you so that you can still use your GC instead.