Yes - I remember years ago when the Catholic schools were private, as in no government funding.
As soon as they accepted that money, it was eventually going to be game over. If you dance to the music, you have to pay the piper.
Governments can make mandates without providing money.
That’s what happened here in Australia.
Catholic schools started to be compelled in the 1950s to toe the government line on certain issues. To begin with, it was an unfunded mandate. They didn’t receive any government funding, but it didn’t stop the government making them follow directives when they wanted to.
Eventually, it was decided here that governments would provide some funding but that came after the mandates.
(The ‘Goulburn School Strike’ was the seminal moment - in response to a government’s unfunded mandate, the Bishop of Goulburn shut down the Catholic schools and told parents to enrol their children at the local government schools - which could not come close to absorbing that many children).
Refusing funding doesn’t guarantee non-interference. If governments want to interfere in private schools, they will do so, whether they provide funding or not.