There is some precedent for this belief. In 1988, courts in New Jersey decided on the now-famous “Baby M” case where they ruled on behalf of a traditional surrogate against the child’s intended parents. Baby M’s traditional surrogate decided to forgo surrogacy compensation and claim the child as her own, though it’s worth noting the child also shared biology with the intended father whose sperm was used.
Gestational surrogacy was not a practice back then, nor were their laws and safeguards in place to make sure these types of disputes were largely avoided. In modern times in states where commercial surrogacy is legal, the surrogate is required to undergo a rigorous psychological evaluation while also proving she is not desperate for money and is financially stable.
The practice of gestational surrogacy vs. traditional surrogacy has also removed the ambiguity of whose child it is as the surrogate is not biologically related to the fetus she carries. Because of the potential for messiness, some states even prohibit traditional surrogacy outright, though with the proper safeguards in place traditional surrogacy can be done ethically and without conflict.
This potential for contested parentage is why surrogates are required to have had their own children whom they are raising or have raised. By undergoing surrogacy, these women are not missing out on their own opportunity to be a parent; they are already parents to their own children.
The vast majority of surrogates do not enter into custody disputes with intended parents. Since commercial surrogacy’s regulation via a wide array of protective laws, contested surrogacy is so uncommon that when it does happen it tends to make local, national, or international news.