The Hillsborough argument is worse than your car analogy. A car is still a car, just a safer one.
Safe standing is not a return to standing on the terraces so what happened at Hillsborough has absolutely nothing to do with the safe standing debate, except as a red herring and excuse not to implement it. With safe standing you would have a ticket to an allocated space. Even if people did try and overcrowd some spaces it would be impossible to generate the same pressures that surging terrace crowds did (it would be more like your crowded tube).
One issue that ought to be raised up front is pricing. Many people think safe standing would be a return to cheap tickets. It would not. I assume the tickets would be less but the club would want to raise at least as much money from the same space. If two people occupy the space of a seat then half price would be revenue neutral, but I think the number would be more 1.5 standing/seat so price drop of a third is the maximum to be expected. And some bright spark in the finance office might observe that there is an extra demand for the standing area and decide that is a reason to charge a premium price for standing or at least justify the same price as for seats. Then the club would gain revenue and it might get more support.
Another issue is for Spurs competitive position. If we go ahead with our stadium it gives us an advantage. While safe standing would allow us to expand capacity further, it would also give other clubs (Liverpool, Everton, Villa) room for more cost effective renovation. Convert some areas to denser safe standing and other areas to less dense corporate seating. That way they can raise revenues through expanding the corporate section and keep overall capacity in the same physical stands.