If I buy a BTC for $30,000, then later decide to sell it for $47,000, Paypal will lose $17,000 if they didn't actually have a BTC somewhere in their vault.

This is the same idea as bucket shop operations that "trade" stocks with their clients without actually processing trades on any exchange.

When prices are rising, with new buyers available, then a bucket shop can do just fine for themselves. As soon as they have more sellers than buyers, at a price higher than they came in, the shop is the one that has to cover the difference.