You absolutely can restrict it. An employer can forbid their employees from taking tips, there are plenty who do so already. There's all sorts of regulation around tipping, and under federal law, employers can require employees to participate in a tip pool or otherwise share their tips with other employees.
That means by extension that the state or federal governments can require employers forbid employees from taking tips. You're free to give anyone you want money in a personal capacity but in a professional capacity their ability to receive it can absolutely be curtailed by the government. There's lots of prior art here, especially around business conflict of interest. You can't tip a government employee for their service.
As a thought exercise, if you go to a restaurant today where tipping isn't permitted, you offer $10 to the server. They refuse, because their employer doesn't permit them to accept tips. What do you do? Force them to take it?
You can choose to go to a different restaurant, you can't choose to deal with another government. Hence the rules should be different.
The rules and regulations of a governmental agency do not need to have any bearing on the rules and regulations of a private business.
The point remains, you can give anyone gifts in a personal capacity, and there are restrictions in a professional capacity.
If I gave my customers a gift it would be an illegal bribe and I could go to jail. My customers are government.
When I was a teenager I worked at a hospital. One of my coworkers was fired for accepting a $1 tip from a patient, we were not allowed to accept any money or gifts from patients or families.
That’s a flawed analogy. The server is not a customer.
"People's freedom to give other people money on their own volition as a personal gift will not be infringed"
There's many situations in life where personal gifts are inappropriate, unwelcome, or illegal. I just gave an example.
You could also make the same argument to defend the "casting couch," yet we seem to have moved past that being defendable.
Could or should are different questions, but you're intentionally reducing the conversation to something much simpler than it really is.