It does take some time to eat something that you don't have a taste for. The best you can do is to realize that hunting requires killing, and that wasting parts of the animal you have an aversion to is just that: a huge waste.
Also understanding that those animal parts are good for you is helpful at a high prefrontal cortex, in the same way that doing something you might not have initially liked, such as exercise, is also good for you, and then after having done it for a while, seen that you now enjoy it. But also understanding it at a visceral, emotional level is key, which is why you're here asking this question.
That said, finding a good way to cook organs in a way to make them more palatable for yourself is the key.
There are other parts to try such as tongue, and heart, which are less offensive than liver or kidneys. (I still don't think I could bring myself to eating kidneys due to the smell.)
Liver is very edible if you have it as either Braunschweiger or Liverwurst, or pate, or if you cook it in various ways to change the flavor, or just mixing it into burgers
Then again, I can't stand the smell of lamb either, but I forced myself to get past that, and noticed that I felt amazingly great (calm, happy) after eating lamb. Probably something in the specific fats, so now I look forward to eating it. I masked it by adding a bit of ghee in the pan before reheating it, and some A1 on top, would help to add some mint, but didn't have any fresh mint at the time.
A few days ago I hid liverwurst in an Omlette (along with broccolli, cauliflower, bacon) and my kids loved it and wanted seconds. My wife made some noises about it, but then she saw me put it in and she's squeamish about liver.
A few months back, I tried heart, but failed at it and overcooked it and it turned out too rubbery. I'll have to find a better recipe for it so as to not overcook it next time.
While it's not the same kind of aversion you referred to, there is another, one I didn't even think I'd ever experience, but it struck me more than just the flavor of liver:
I recently bought bison tongue and wanted to cook it, but once thawed, i looked at it and immediately realized that this was attached to a living creature and felt a bit weird about it - I saw a playful buffalo running around in the grass, and using that same tongue to taste the grass it ate... But I worked through that "vision" by realizing that this animal was already gone, and the best thing I could do is not be sad for it, but rather honor it by not letting its tongue go to waste.
I think we're too used to muscle meats and too removed from life itself to fully understand the cycle of life at every level. Sure, at the highly removed, frontal cortex, we understand what meat is, but maybe not at a visceral level.
So I worked through that, went ahead and cooked it anyway, and had it with mustard and appreciated all that the animal was.
While I didn't exactly intend on deceiving the wife into thinking this wasn't corned beef, she just assumed it was and I just nodded adding it's actually bison, not beef. I think I'll tell her today it was actually tongue than muscle, but I can already hear her reaction. :)
It didn't exactly taste like corned beef, but it was very close. It certainly was a lot tastier than liver, and far too tender. It needed to be drier, and I should have used a lot more salt than I used. Maybe I can rescue this by salting and pan frying the slices in ghee.
I should have followed this recipe. instead of attempting it up on my own.