Niyoga as a process was apadharma process. Something that is acceptable only in the most dire circumstance.
Niyoga process mentioned for any charachter in Purana have only 2 instances where the man doing Niyoga was a relative...
Vyasa (BIL) for Ambika Ambalika and Rishyashringa (SIL) for Dasrath's wives (that was for sure Niyoga covered up as yagya Prasad).
Even these two were eminent Rishis who would never ever try to take over the kingdom or claim the child and it's inheritance.
All the other cases of Niyoga is non relative Rishi or a Deva. Eg. Vashishtha for Rama's ancestor or Dirghatamas for Bali's wife
Even those women who were widowed childless by Parshuram went to eminent non relative Brahmana for Niyoga.
There was no way any of the kings would ever be allowed to do Niyoga, be it Shalya or his friends. Same way if Satyavati had no Vyasa, Bahlika will never be called to do Niyoga. Infact remarriage was offered to Satyavati by Ugrayudha and Bhishma killed him to protect Hastinapur. Offer of Ugrayudha meant his and Satyavati's son would inherit the throne.
Widow remarriage did happen but Niyoga was not widow remarriage. It was a one time sperm donation where donor had to be somebody whose sperm had value but who would never take over the inheritance of child so born
So Madri's relative or family member would never be accepted.
Pandu and council of Hastinapur would only accept a Rishi or someone better than Rishi - a Deva.
Now this Deva / Indra could be either of two things.
1. Actual king from somewhere in Tajikistan who was at that point acknowledged as sovereign ruler of all earth. And everyone paid tributes to him ( other guys his council). Maybe he and Devas owed Durvasa and helped Kunti to oblige him
2. Anonymous guys whom Kunti approached and got the job done. While Durvasa may have given a blessing kind of thing that if you pray to a particular God you will get a son like him (not son from him) that Kunti converted to a propaganda with Pandu's and Durvasa's help that I summoned Deva and he gave me the child.
I am more inclined on the second as Kunti would have already worked this propaganda part in case Karna is ever found out to be her child. He was a sword hanging on her reputation so whole Durvasa Mantra thing was to be used to salvage her chastity. Durvasa may have been the one to suggest it to protect her out of pity on her plight and because she served him with devotion.
So Durvasa basically gave her a kind of birth certificate with bio father's name as a particular Deva and mother name as her, that Kunti filled in with a Deva name of her choice after niyoga
Since she had the birth certificates in her name, Madri could only get it of Kunti would give it to her
And Durvasa had enough clout for his testimonial to be accepted at face value by other Rishis and kings.
BTW it's interesting that neither Kunti nor Madri summon Agni.
Logic dictates that Madri should have summoned Agni or Surya or even Varuna as Ashwini are too far down the pole to be a good choice. Ashwini are not known for their power or prowess. Only for their beauty and medicinal skills.
But we have Draupadi as daughter of Agni marrying them.
Karna already had Surya as father, only Varuna got left out.