Why would the "Catholic one" suppose that God "probably would not want a sinner in that role (the mother of the incarnate Son of God)"? Was not one of the main intents of the incarnation that Jesus was:
For what the Law could not do, weak as it was through the flesh, God did: sending His own Son in the likeness of sinful flesh and as an offering for sin, He condemned sin in the flesh, so that the requirement of the Law might be fulfilled in us, who do not walk according to the flesh but according to the Spirit. (Romans 8:3-4)
And:
Since the children have flesh and blood, he too shared in their humanity so that by his death he might break the power of him who holds the power of deaththat is, the devil and free those who all their lives were held in slavery by their fear of death. For surely it is not angels he helps, but Abrahams descendants. For this reason he had to be made like them, fully human in every way, in order that he might become a merciful and faithful high priest in service to God, and that he might make atonement for the sins of the people. Because he himself suffered when he was tempted, he is able to help those who are being tempted. (Hebrews 2:14-18)
There WAS no need for a "sinless" mother in order for the Christ to be born and be without sin so that He can make atonement for the sins of the world. That Jesus is God incarnate is how he could be sinless - it had nothing to do with His mother also needing to be. Mary was a human woman, born under the law, born under sin, and her faith in the Savior Jesus Christ is what saved her and cleansed her from all her sins just as He does for us.
That really IS the problem, Catholicism decides what is doctrine whether or not God's sacred word states it is so. I believe they have it exactly backwards. God tells us what is truth and our response is to believe it and obey. We don't get to make it up as we go along and superficially search for verification from Scripture afterward.
There is a difference between where Christ is from and whom He saves. That He mingled with sinners is very true; but His sinless mother indicated that He Himself is from where there is no sin.
Catholicism decides what is doctrine
As simple as that. True.