re: Why do people think “paying cash” gets them deals?

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Trading all but guarantees someone is going to be getting beat.

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Maybe. But it goes the same for cash. You can trade a dollar amount (subject to inflation...) for an item you want. You certainly don't want to pay more than it is worth. How much the item is worth depends on how bad you want the item, and how many people also want the item, and how many of that item are available (supply/demand). Also depends how many dollars you have. I doubt a billionaire haggles much.

In all of this discussion, I believe you'd agree, the value of the items being traded (be they dollars or actual items) are based on who has what and who wants what you have.

I get your point. My contention is the value equilibrium of a trade is ultimately decided by the satisfaction of the parties involved, not the "potential value" of the items.