One of the benefits of PayPal for consumers is that they don't have to share billing information with all the different merchants they're buying product from. As a protection thing, PayPal doesn't share "billing" information. The only address they will provide is a shipping address when necessary.
Like you said, though, you could simply make sure the shipping address provided was "confirmed" which means it matches the billing address of a credit card the buyer has on file with PayPal.
True billing addresses only come into play with PayPal when you're using Payments Advanced or Payments Pro and you're working with actual AVS and CVV2 results.