In a few cases, the chip in my chip card (or maybe the chip reader) has malfunctioned. After inserting the card and failing 3 times, it said to swipe the card instead. What's preventing me from cloning the magstripe to a card with a broken chip and then bypassing the chip security entirely?
As others have said: nothing. The idea is that the magstripe is a fallback that will be phased out completely as chip cards become more widespread.
Change takes time.
There's nothing that prevents you from breaking the chip, or even just deliberately misusing it (i.e. putting it in upside down) to make the terminal think it's broken. However, every time a chip card is swiped at an EMV-capable terminal, the POS needs to tell the card networks why it allowed that card to be swiped - otherwise, the merchant will get hit by some pretty hefty fees from Visa and MasterCard.
There are two legitimate reasons for a fallback - either the terminal and card can't agree on a payment brand, or there could be something physically wrong with the chip. If your bank starts receiving lots of indications that your chip isn't working, they may simply just send you a new card. After all, you're clearly more vulnerable to fraud now, and they don't like that.