I think we’d reject as a matter of morality and equity that the prettiest people, the people with the best story, or the ones who can pay the most, should get access to this very scarce resource.
The prettiest and wealthiest people already receive preferential access to every other scarce resource. In particular, the wealthy can afford medical care vastly superior to that available to the poor. How is a new kidney so different from a new cancer drug?
Looking at the issue from another perspective, why is the outright, informed, deliberate sale of organs by a living person so unthinkable? People already sell their time, and often their health, through their jobs. Black lung, anyone? Experimental subjects are compensated for Phase I clinical trials, which assess the safety rather than the efficacy of a new drug. Unfortunately, I've been unable to find any sites advertising the rates for experimental subjects. Perhaps because of the aforementioned dubious ethics of allowing people to sell their bodies, very little information is publicly accessible, although you can request a quote for the value of your health.
If I were starving and homeless, I think an offer to barter a kidney for a year's worth of food and shelter would seem more than fair. Why should the government be able to tell me what I can and cannot do with my body? In this light, the selling of organs seems rather similar to prostitution or assisted suicide. On the surface, the government appears to be protecting the poor (in the case of prostitution, and the godless heathens, in the case of assisted suicide) from exploitation, but I would argue that poverty itself is unjust in a land of plenty. How is it ethical to strip people of one of the tools with which they might free themselves from poverty?