The bribe isn’t for them, its for you.
So, I am bribed by paying them after I die?
I suppose that might matter in some cases. I’m pretty sure I won’t be using my kidneys after I’m dead, and my relatives already know I am favor of organ donations. Alas, older organs aren’t that valuable, even to the desperate, so I will probably age out of eligibility to donate.
And that’s why we are likely to continue to have shortages. Most useful donated organs come from young adults killed in auto accidents. And thankfully, there aren’t a lot of those. Most of the things that kill us (including old age) also damage our organs.
This has been my experience as well. When you renew your license, they simply ask if you want to or want to continue being an organ donor. If yes, it’s printed on the ID, no sticker.