Does a V5C prove ownership?
No — a V5C proves who the registered keeper is, not who legally owns the car. These are often the same person, but not always: for example, if a car is bought on finance, the finance company legally owns it until the agreement is settled, even though the person driving it is the registered keeper on the V5C.
The DVLA states this directly on the document itself. For proof of legal ownership, you'd typically need a receipt, invoice, or finance settlement confirmation — which is also why checking for outstanding finance matters, since it affects who actually owns the car you're buying.