You’re absolutely correct.
But I think what the OP is saying is why does a Catholic “Have to” have the child baptized in Church as the ‘norm’, whereas supposedly nonCatholic Christians can baptize anywhere (although even there, I think the majority of nonCatholic Christians have baptisms in their churches, or the baptisms might be done in a river but are done by the pastor, not by their parents.)
In any case, the NORM is for a Catholic child to be baptized in a Catholic Church.
And even if the OP’s ‘nonCatholic Christian parents’ ‘started their own church’, well, that would make the parents (one of them anyway) the pastor of the church, right? And even if the church was in their home, it would still be technically a church, by their definition, right?
So the baptism would still be done by a leader of a church, in a church. . .
Not all that different from Catholic baptism after all.