The dividend is paid to the owner of the share because they own...

The dividend is paid to the owner of the share because they own it. The shorter has only borrowed the share from the owner and has no ownership thus has no entitlement.

Lets say I lend you a car to rent.

I still have to pay all the costs and receive all the benefits of owning the car.

All you have to pay is the rent to use the car as you see fit. If you happen to use the car to make or lose money all that I require you to do is give me back the car that I lent you in the first place.