Then other parts of your body would start failing in interesting ways. Interesting from a scientific perspective, but I'd imagine pretty horrible from a human one. Your heart and kidneys don't fail by themselves, your entire body suffers cumulative damage from mistreatment and it's just those parts that fail in a terminal way first.
Right, but oftentimes (but not always) the "cumulative damage" is causally linked to a single source. For example, when your kidneys fail, the rest of your body suffers crazy loads, since the kidneys are tasked with maintaining electrolytic balances in your blood, which other organs depend on. For example, failing kidneys don't clean out potassium from your bloodstream, which causes your heart to come under increased load and usually results in sudden cardiac death.
So replacing a failing kidney early on will keep your heart in a much better shape. Of course, in the long run, you'd need to replace everything, and in the very long run there's always cancer...
That is not necessarily true. There are lots of instances where a specific organ is experiencing failure and could be replaced and the rest of you is pretty normal functioning. Especially in kidney failures or injuries, liver failures or injuries, and hearts as well. Think about all the people on dialysis. Lots of them are awaiting a new kidney from a donor. What if, instead of waiting 2 years for a match they grow you a new one in a few months? Same thing with the incredibly regenerative liver. Hearts might be another story, not sure there. We'd probably need to be able to grow those completely outside the body first.
I wonder how far you need to go before a brain transplant is cheaper / safer than replacing it. I can see replacing major veins as being doable but capillary's are far less so.
Sure, we can; but if we'd thought that way before we would never have developed medicine. For some reason in every generation there are people who seem to think that using the medical science they are familiar with (or just slightly less) is okay and normal, but using anything more is somehow contrary to nature. But using our mind to develop new ways to overcome the limitations of our bodies is our nature just as much as the failures of our bodies are.