As your link says, they are not allowed to pay less than minimum wage. They are allowed to set hourly wages for tipped employees lower than minimum wage, but they have to ensure that the total compensation of the employee averages out to minimum wage.

Realistically, if the restaurant owner has to contribute anything beyond their hourly rate to the paycheck of the server, that server will be fired (or the business is so slow that it will go under soon).

Not necessarily. I briefly worked at a 24x7 roadside place where tipped employees would occasionally fall under minimum wage due to slow periods or customers who would either stiff the waitress or tip out $0.35 on an egg, toast, coffee which cost $2.65.

That place systematically robbed those staff by refusing to pay minimum wage and falsifying records and withholding taxes based in nonexistent tips. The Labor department fined then pretty heavily -- but that isn't an uncommon practice.

I'm not sure how this contradicts what I said. Your experience showed that the business was not contributing anymore to your paycheck than the normal wage. Had they not been breaking the law, they probably would have fired someone due to the "extra" expense.

My experience as a server was a lot of < $1 paystubs after taxes on tips.

Exactly - so any tip just reduces how much the owner has to pay to cover minimum wage, making it a subsidy to them, not to the server, who gets minimum wage either way.

It's not really a subsidy though, it's just a complication of the payment.

It could be a subsidy if it wasn't an established custom to tip, but people know quite well that they are paying the server a tip for the service.

I suppose the fact that it enables the restaurant to under-report wages and thus under-pay wage taxes is a bit of a subsidy (but then some of that savings probably shows up for the customer).

It is a subsidy, because the server gets the same, it's the restaurant that pockets the difference.

Say the server works 40h/week like a regular full-time worker, so the minimum take-home pay under federal law is $15000. If the server gets $0 in tips, the restaurant must pay the $15000. But if the server gets, say, $5000 in tips, the restaurant can pay only $10000.

The server gets the same money, only the restaurant gets to save some.

Right. In practice, servers get $15000+ in tips. So there usually isn't a subsidy.

It also has a non zero impact on the menu prices (they would go up if tipping wasn't expected)