You're both wrong -- if you need two different resources as inputs to make something, and either missing makes the process impossible, it doesn't matter what your subjective opinions are when deciding which is more valuable. They are equally necessary and that's all there is to it.

So how come some input resources are priced a lot more highly than others? How come sometimes, say, labor gets a big cut and capital a small cut, and other times the other way around?

Supply and demand. Bargaining power. That's why. Nothing to do whatsoever with the actual value involved in the process.

What portion of the value we apportion to the suppliers of each resource is purely a political question based on bargaining power -- who is less willing to withdraw from providing their resources, etc

To use your sports analogy, if this magic genius coach can easily find plenty of any old players to execute his strategy with, then I guess he will manage to absorb the lion's share of profits. If it's hard to find players relative to supply/demand, then it doesn't matter if they're dipshits who barely manage to do anything and who just do what the coach says, it doesn't matter if anyone could do what they do. All that matters is the coach needs them and there's not enough of them. You can't have a team without the coach and you can't have a team without the players, regardless of their respective merits.

How you divide the spoils among resourceholders all of whom are critical to the process is purely political, based on leverage and negotiation, not based on actual "value" which cannot be computed unless a sports team can exist without players or without a coach and merely be a little, measurably worse.