> at the risk of enticing some folks, who have the means but love a deal, to be penny-wise pound-foolish at everyone's expense including their own

If no one did this, wouldn't that just mean that the price discrepancy kept rising and rising?

That's an interesting point! For simplicity, let's divide into 3 segments: poor, middle, rich. Poor can only pay with coupons, rich can't be bothered/seen and only pay full price, middle can pay in full but some of them decide to use coupons because it's deemed worthwhile (so really 4 segments). Suppose middle doesn't exist, and prices are set in a way where tons of poor people are fed while also maximizing profit from rich people; perhaps this means giving the poor a 50% discount.

Now introduce the middle. The portion of them paying with coupons is anywhere from 0% to 100%. The restaurant makes massive profits from them at 0%, and makes no profits from them (but covers all costs) at 100%. Let's say it starts near 0% because initially nobody knows about coupons and over time rises near 100% once the information has spread around. Profits decrease so something must be done to counter that. Shallower discounts (say, 25% off instead of 50% off) is a possibility, keeping the discrepancy tight, as you suggest. But increasing menu prices, while leaving the huge discrepancy, is also a possibility!

Either method will get the profits to the desired level. But only one method -- maintaining the large discrepancy -- sustains the goal of maximizing the number of poor people fed. Therefore, a restaurant with that goal will choose that method; a restaurant without that goal simply won't issue coupons. The discrepancy, if one exists, is therefore constant regardless of how many penny-wise pound-foolish people are in the middle.