I'm a bartender. Half my paycheck is tips (and still isn't enough to live in the area, but that's another issue). There shouldn't be a tipped minimum wage. Period. Tips should be optional, not a way for employers to make customers pay their employees' wages.
Long gone are the days when you made $7.25 an hour on your paycheck and took home hundreds in cash per night. People are tipping less, eating out less, and carrying less cash. Paying everyone a livable wage and eliminating the need to tip would help everyone, except the owner class.
It depends on the establishment. Some places will ask you to report legitimately, while some will let you work off the books illegally and not even record you as an employee. My last boss wanted realistic numbers, so I put in about 80% of my tips.
I guess I knew people in the 90's when everyone paid cash. I knew one bartender who had a stack of cheques she never got around to depositing because she was rolling in cash.
When I delivered pizza as a side job back in the early 90s, I asked my manager how I should handle reporting tips and he said "Go ask James, he can tell you how it works." James was their senior driver, and a long-time friend from school, and he said, "I don't know about the other drivers, but I have never received any tips." I got the message. This was back before online ordering was a thing, and if somebody wanted to pay by credit card they would have to come into the store, so it was 99% cash. Every now and then a regular customer would pay by check, but the check would be made out to the company, so I would just take all my tips out in cash when I settled up at the end of the night.
The minimum wage for service staff in the US is bonkers, and a great example of 'if they could pay you less, they would'. I don't know how often employers actually pay above 2.13 when they don't have to, but I've read a lot more about being paid 2.13 plus tips than above that.
Not taxing tips is actually a gift to the employers more than to the employees. When a tipped worker claims 100$ they will have taxes and SSI withheld from their hourly wage AND the employer will also have to pay into SSI for that claimed tip amount.
Do away with taxes on tips, then make the argument that there shouldn't be a minimum for service workers at all. Eventually we get rid of minimum wage completely and SSI.