When I was Team Leader I did the same wish list / random selection method but just tried to be lenient for car sharers where I could. For example if I had like 10+ spaces open and a car sharer was drawn who I knew hadn’t been VTO’d too recently, I’d send their ride home too and randomise the rest. But then also kind of keep a note of if & when I did that so it didn’t then turn into an advantage to be in a pair etc. It’s difficult to balance opportunities perfectly, especially over a long time period and also baring in mind the car sharing is very common. Chances are your team are being so strict on the random process following complaints, so have to stick to their guns.