It's really not that most restaurants wouldn't make up the wages to the minimum. The minimum wage in most US states is not for most people would call a "living wage, it's around $15k/yr. It's not uncommon in a major city for waiters to be able to pull in $30-60k/year. The thing is that in many place tips are paid out in cash at the end of their shift, so if you don't tip you're literally taking money out of their pocket for that day.

Many paycheck I've seen from most wait staff is less than $20 for two weeks. The employer is required to withhold taxes based on the hours worked and tips reported, so the whole paycheck basically goes to taxes.

As for cases where someone doesn't make enough tips to hit minimum wage, in many places not making enough tips is a sign that the person isn't doing a good enough job. So they won't be fired for "asking for minimum wage", but because "their performance was unsatisfactory".

So yes, in practice most wait staff do live off their tips. Depending on the locale an extra $10 or $100 in tips could be the difference in what their kid eats that night or whether they can make their car payment tomorrow.

(Personal story time: I will always remember helping my mom wait tables at a local diner when I was young and someone tipped $10 on a ~$10 meal and instead of eating hamburger helper that night we got to go out an eat at Long John Silver's.)