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The problem with economic theory in general is that it assumes the consumer is "rational", that is, they make the best decision based on the available data. In reality, individuals make emotional decisions and behave irrationally.
All things being equal, paying the highest interest debt obviously makes the most sense. Given that many people lack the motivation to exercise financial discipline, paying the lowest debt may be the best strategy for some.
Ramsey's perspective is mostly informed by his unfortunate experience in over-leveraging credit (too much borrowed money), having his debts called (banks requiring debt be paid off in short notice), and ultimately losing his investment properties when he couldn't pay. There is risk of a domino effect when leveraging debt because 1 lender can feel uneasy, call their debt, and that in turn triggers the other lenders to call their debts. It's the opposite of a bank run.
That said, most people aren't financially literate, and even worse, have no interest in becoming financially literate. Ramsey has strategies that uninterested debtors might find interesting enough to practice, and improve their situation.
Mr. Money Mustache is for people who are interested in financial literacy.
As an aside, I've never paid interest on a CC even though I primarily pay with them. I was putting $10,000 on a CC every 3 months to pay my wife's tuition, and banking the 2% reward.
There is a strategy for investing credit card debt called the "App-o-rama". The basic idea is that a person applies for many credit cards at once that offer no interest terms and/or cash incentives. During the no interest period, they monetize the credit and place the money in a safe, interest-bearing account such as a rewards checking account. At the end of the free interest period, the card is paid off in full, the investment interest is kept, and the process is repeated.
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