I've done one pickup near Waverley with the rest at Turnhouse, and I definitely recommend the airport option. Edinburgh developed long before automobiles so the streets are terrible for driving, and the city leadership has only adopted policies to discourage driving further. In fact, if you're only in Edinburgh for one night, I might suggest just staying out at the airport and taking the tram in and out if you want vastly better dining options.

Also agree with the advice regarding the vehicle you rent. Plan on packing your vehicle to the gills; taking something much larger than you need will only hurt you at the gas pump and when trying to find available parking.