You can prevent a lot of fraud by simply requiring both a postal code and the CVV on the card. It's unlikely that a card thief has both pieces of data. (If you steal the physical card, you won't have the postal code, if you buy a database of cards, unlikely that they will have CVV's)

I wish this were true. We have both AVS and CVV checks on payments and things still made it through a property rental advertising service we run. It was an uphill battle until we implemented https://www.maxmind.com/en/minfraud-services

This isn't even remotely true. There's tens of thousands listed for sale for < 10 USD all over the web right now.

But not all countries use postal codes, and checking of card registered address details is always a bit fuzzy simply because there may be several correct but different ways a legitimate cardholder might enter them.