This is what gets brushed aside when the organ donation process is activated. Only now do we have results from long term study of donors and some of them as pointed out by the author are not good at all.
This problem is not confined to kidney donations for instance, is a growing body of evidence that points to long term complications from another medical procedure - laser surgery (lasix). Here many patients suffer degenerative vitreo-reinal changes over time (if you can, avoid it).
It is not actually the case that donors do not benefit from the current process. Outside the US in the largest market for kidney surgeries India, they are paid; as well as in the Thailand and the middle east. End stage renal disease is one of the worst conditions to die from, and everyone involved with an afflicted patient is desperate for a solution since they suffer almost as much as him/her.
So,No. Paying for organs will not solve this particular problem. But neither will doing nothing. One solution that can improve the number of donor organs is to make it mandatory that organs from qualified people who die violent or accidental deaths be immediately considered for transplant to those who need them. But you can guess this proposal would be (illogically imo) opposed in many countries.
Edit: On a related note, I have been mulling over an idea for a while for a startup to reduce the cost of dialysis for patients with renal disease. At present, it is prohibitively expensive and some patients require 2 sessions a week. If anyone is interested in joining me, reach out. My email address is in my profile.