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You need to borrow $200 for 60 days. Is it cheaper to do so for a flat $10 fee on repayment or by using a credit card with an APR of 20% (compounded monthly)?

I can't do the math to answer this question in my head. I suspect you can't either. The point of normalizing the cost of borrowing money to an APR is so that a consumer can make this decision without having to solve exponential equations.

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If I had a credit card with a limit large enough for the need, I wouldn't be even thinking about payday loans in the first place.

This hypothetical doesn't happen in the real world; there's no time where even the most awful subprime credit card (even at cash advance rates!) will be cheaper than a payday loan.

Its usefulness as a metric for a loan intended to be held for a couple of weeks to a month is very limited. Maybe when shopping across short-term lenders, but at that point, it's more intuitive to think in terms of how much the fee is in absolute terms.

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