Yes, I have a ton of experience in this. In most states, the issues of custody, visitation, and child support are each a separate issue. In our case, I continued to court each time my child's co-genetic contributor wanted a review of the ordered support he didn't pay. In each hearing, he was unable to prove that he could not sustain a reasonable income in his field and was ordered increases. Every.single.time.
That didn't stop him. He dragged me into court no less than 3 times a year for years. All the while, he never exercised his rights of visitation and only proved to be a hindrance to conservatorship of basics like medical care.
By the time the child was 18, I had gained full conservatorship and $60k in arrears owed in child support. If it's ever paid, that amount will go nicely toward higher education or a home down payment for the child.
I knew the co-genetic contributor would probably have signed off his "rights" gladly if he had the option. I never could do that to my child. My child has the rights to financial gain if ever paid and to the knowledge that I advocated for them all those years. If the co-genetic contributor passes away or if a windfall of inheritance ever came his way, or if he files for SSI, my child has rights to that as well.