The problem is that the MID/MCC of the store you shop at won't be passed-through to your real credit card (as PayPal's the one charging that card, not the store

The problem is that the MID/MCC of the store you shop at won't be passed-through to your real credit card (as PayPal's the one charging that card, not the store). You wouldn't get category-specific rewards (5% on groceries, 3% on gas, etc), which is one reason people carry multiple cards on them in the first place.