> Yes, even if the second group of those not operated on would die, I'd still not "allow" someone to die so I could harvest their organs.

Okay. That is what most people want to say about that case, too.

Then the problem becomes... How we can justify killing the one to save the six in the first case... While simultaneously justifying killing the six to save the one in the second case...

> People are going to die all the time that I can't save; in your moral world, I'm only going to save those I "personally" can with my skills alone.

Yes. But in both the above cases you have two options within your power:
Kill the one to save the six
Kill the six to save the one

Most people say 'greatest good for the greatest number' to justify their response to the first case.

But 'greatest good for the greatest number' doesn't justify their response to the second case. So... Something else seems to be going on in the second case. And what is going on in the second case doesn't seem to be going on in the first case.

Or maybe... We should change our judgement of one of the above cases such that they come into line?

(That is the problem that a lot of ethical theorists are worrying about)