These guys “get it” and are very good at what they do.

Growth is dependent on several factors.
1. Primarily, expectations of the future. Consider a baseline agricultural society. If, as a farmer, you foresee a poor harvest, you are not going to stock up and take risks with your crop or livestock. You’re going to make sure you limit risk and bring in as much as you can. This is the key to growth – an optimistic outlook for the future, the willingness to engage risk because reward is either well known or expected to be worth it.
2. Ability to leverage current or future surplus. Essentially, ability to take on debt. While this is not essential for growth, it’s huge engine for smaller economies as they foresee good times and take advantage of what they currently have to push for more. Leverage means the payoff can be lower for optimistic expectations because there is less at “risk” (though push this too far and you’ll undermine yourself).
3. Innovation. This single thing can create massive growth because it can ignite #1, but even if it doesn’t, it vastly changes the way things are viewed. The yoke was an amazing invention because suddenly huge amounts of land could be plowed quickly and easily, with little human effort. Imagine if you’re a farmer, you invent the yoke and your neighbor is still working by hand. Your view of the world will be tremendously optimistic.
4. Productivity. This is a side effect of optimism and/or innovation. Productivity means MORE for LESS. Which means more people benefit from the gains of growth. Luddites used to smash looms believing they cost people jobs and livelyhoods. Today, it’s clear the Luddite philosophy is backward thinking, as we employ more people at higher wages, with better livelyhoods than ever before. Still, we have many Luddites in the political class…..

So where does cash come in? Primarily in point 2. But as I said, it’s not essential. There can be growth in a barter economy, and indeed there is. There can be growth in an economy with salt as its currency (as it once was). There can be growth in an economy with gold as its base. But the reason fiat currencies work (as long as they are well managed and moderated) is that they help to grease the cogs of the leverage/risk engine. When gold was the basis of the economy, innovation found it hard to take root because growth was based on monetary velocity, which has limitations. With a fiat currency, you can have monetary velocity increase AND increase the amount of currency in circulation – much more enabling (if inflationary at times).

Would the economy prosper or faulter if Americans only paid cash for things?

My initial thought is that the economy would spiral downward (much like it is right now). I am assuming of course that most people will spend all of their “disposable” income on paying off their non-mortgage debt. Eventually the economy would bottom out when Amercians finally had enough in savings to start spending again.