So, it seems you disagree with the premise that the most basic purpose of a tip in the US system is to provide the customer's share of the value of the the tipped employee's work. You can disagree, but I say the IRS rules imply otherwise. It can only be called a "gratuity" with a wink. As I see it, a tip is not a reward on top of a worker's full compensation; it's an integral part of their compensation.
You're correct that the employer is supposed to make up the difference, and I probably should have stated that up front. What's missing from the calculus here is that the "regular minimum wage" is not necessarily the whole value of the tipped employee's work. But the employees's wages plus, say, 20% should in theory add up to what their work is worth. The employee's wages may be anywhere from the tipped minimum wage to way above the regular minimum wage, but the employer tries to set that wage so that when the customer's 20% is added, the total is what the employee should be compensated for their work. If it's too low, the employer risks the employee leaving for higher paying work.