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They do get priority. But there is still issue of compatibility. And one still needs to take a lifetime of medicines. I myself found my kidneys were failing recently and did a lot of research.

Came here to post exactly that question/suggestion.

I could imagine that it might also be "and can put one blood relative or immediate family member into a high-priority category" to encourage donation/control for the "yeah, but I'm saving mine in case my kid needs one" objection.

..... or get your kidney back when the person you donated it to dies

Not a doctor, but I'm guessing that it's hard to re-transplant one and you have to get the person to die the right way such that any donation is even possible.

Seems like the way to solve this, without opening up market pricing for them. In theory, this would create a marketplace where supply outstrips demand.