Actually, to the best of my knowledge no Bitcoin debit card is issued by a bank, or at least, doesn't need to be. Visa and Mastercard work with businesses as long as they fill certain criteria and authorize them to be able to issue cards.
In fact, many of the places that you get the cards from, are not actually the ones who physically issue the cards. The Xapo card is just one example. It's issued from the Caymen islands/Gibralter (if I remember correctly) by another company for the companies that send us the cards. I think at least one company issues cards from Hong Kong.
Basically, the actual card issuing company is usually in a region that has fairly flexible tax laws/economic growth incentives.
As to why send Bitcoin to a debit card, well that depends on the person. Some may make profits from trading and then wish to withdraw in fiat. For me personally, it's just so I have the flexibility to do so should I need to.
There's also another reason: Say you bought one BTC at $280, and then next week the price shoots up to $1000, you can then deposit it to a debit card and lock it in as fiat at $1000. So no matter if the price does go down, you now have that $1000 one way or another.