A good thing funded by taxes on a bad thing is pretty common in America.
Or would you prefer the oil and gas revenues goes into the pockets of your local oligarch?
In a U.S. First, New Mexico Opens Doors to Free Child Care for All (2025)
The inevitable and unfortunate result of "handing out 'free' stuff" however will always be widespread fraud.
Will there be "some" fraud? Sure, but unless your system is really broken it will be a drop in the ocean. Some abuse is the price of any universal thing. E.g. you all have public roads. Most people use public roads in spec. Some people abuse them, do burnouts, run overweight trucks over them, etc. Is that a reason to remove public roads? No. It is a reason to police them.
Some people throw trash into public parks, is that a reason to get rid of them? No.
Anything nice you will ever have will be treated unfairly by some. Taking that as an argument to not have it just makes sure you never have anything nice tho.
The optimal amount of fraud is not "zero".
The goal is to enforce just enough to keep the fraud below a threshold while not making it too onerous for the people who need and receive the benefit.
And, if you really want to go after fraud, you are far better going after the people who commit it at an organized institutional level: see GOP leadership in Florida and Medicare.