What all this means is she could "not name the father" if she wanted. At the time of when she could have placed Gabriel with an agency or adoption professional in SA, she could have lied and they would not know any difference and that would make it legal in the eyes of the agency or adoption professional.
I don't think that's the way it works. Even if she claimed not to know who the father was she would have to show proof of having looked for him. Also, even if the adoption agency took her word for it, the adoption would not be legal in the eyes of the court when it got to court. Now, if she dealt with an illegal adoption agency that specializes in bypassing the courts, then sure, but that doesn't make it legal.
The putative father registry that you mention is a possibility for fathers who want to assert their rights, but not a requirement.
Here is a FAQ about fathers' rights in TX from
http://iaatp.com/docs/FAQs-TX.pdf
BIRTH FATHER RELATED
1. Does the father of the baby have to be notified of the birth and the mothers adoption plan?
 Yes.
The birthfather must be notified of birthmothers plans to place baby for adoption, either personally, by service or publication. Reasonable efforts must be made to locate the father. In Texas, there is a paternity registry where a putative father
can assert paternity.
 Registry/Paternity Requirements to Receive Notice Statute: Fam. Code
§§160.402(a); 160.312
 The putative father may file before the child's birth, but no later
than the 31st day after the child's birth.
 Registration requires a completed Bureau of Vital Statistics form,
signed and acknowledged by the putative father.
 § 161.002. Termination of the Rights of an Alleged Biological Father
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
Termination of parental rights may be taken of the alleged father
only if all efforts have been made to the satisfaction of the court to
engage the father in planning and or waiving his rights.
2. How is the father of the baby notified of the mothers adoption plan?
 The father of the baby is notified of the adoption plan through personal
contact, service or publication.
3. Can the father of the baby choose the adoptive parents?
 This is not addressed in statute.
 Birthfather and birthmother are able to choose the adoptive parents.
4.
What happens if the biological father of the baby is not the babys legal
father?
 Rights of the biological father of the baby must be terminated regardless of
whether or not he is the legal father. Refer to above statute.
5. What happens if the husband (legal father) is not the biological father?
 The legal father must sign a waiver of interest and the putative father asserts
his paternity via the registry.
6. What happens if the mother does not know who the biological father is?
 § 161.107. Missing Parent or Relative
(e) The department shall be required to provide evidence to the court to
show what actions were taken by the department in making a diligent
effort to locate the missing parent and relative of the missing parent.
7.
What happens if the mother refuses to give the name of the biological father?
 Diligent efforts must be made to determine the name and whereabouts of the
biological father.8. Can the biological father of the baby give notice that he intends to plan for
the baby before the baby is born?
 The birthfather can work on an adoption plan at any time during the
pregnancy.
9. Can the birth father change his mind before signing the legal consents to the adoption?
 Yes, the birthfather can change his mind at any time before signing legal
documents.
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10. Can the birth father communicate with the adoptive parents and child after the adoption?
 Texas law does not restrict contact nor does it provide for a legally binding
contact. Common practice involves trust agreements.
11. Can a minor father independently sign legal documents placing the child for adoption (consent to the adoption)?
 Yes.
12. Can the birth father change his mind after signing the legal consents to the adoption?
 In general the answer is no, the father cannot change his mind after he has signed the consent/relinquishment/waiver for adoption purposes.
 § 162.103 (10)(e) The relinquishment in an affidavit that designates the
Department of Family and Protective Services or a licensed child-placing
agency to serve as the managing conservator is irrevocable.
 § 162.1035 An affidavit of relinquishment of parental rights or affidavit of
waiver of interest in a child that fails to state that the relinquishment is
irrevocable for a stated time is:
1) revocable only if the revocation is made before the 11th day after the
date the affidavit is executed; and
(2) irrevocable on or after the 11th day after the date the affidavit is
executed.
13. Can a birth father anonymously surrender a newborn child?
 Yes
14. What happens if the birth parents do not agree that adoption should be the option?
 Adoption requires consent from both birthparents. If, birthfather is
unavailable, the process can only move forward within the boundaries of the
law i.e. service, publication, or searching on paternity registry for birthfather