This topic is one of the most misunderstood areas of Catholic theology: not only by Protestants, but also by not a few Catholics.
Human beings cannot fully understand many complex and deep areas of theology, because they are ultimately mysteries held in faith. But we can rationally understand a lot.
In becoming man, God chose to involve a human being, Mary. God didn't have to do so. But He chose to “include” Mary and human reproductive biology Mary was a real mother, not just a biological “conduit,” so to speak.
The Church fathers widely taught that Mary was the “Second Eve.” Eve said “no” to God and so caused (along with Adam, and indeed, all of us “in” them: 1 Cor 15:22) the fall of man. Mary, on the other hand, said “yes” to God at the Annunciation and so helped bring about (in an essentially lesser, and non-necessary, non-sufficient way) the redemption of humanity, that was brought about by the work of Christ on the cross (Whom she also freely offered up to God the Father, just as all Catholics do at every Mass).
We “receive” Christ, the incarnate God-Man with a human nature, through Mary. He received His human body from her. Hence we honor Mary above all other creatures as the Theotokos – the “God-bearer”.
Mary is Mediatrix in that way, but she is in a second sense also. God clearly uses many human beings as mediators. We pray for each other. Moses interceded and “atoned” for the Jews in the wilderness, and God decided not to destroy them (Ex 32:30).
If Moses could successfully intercede on behalf of an entire sinful and disobedient group, and if Abraham's prayer could spare his nephew Lot (and potentially Sodom and Gomorrah also, if enough righteous men had been found there: Gen 18:20-32), why is it so remarkable that God would choose to involve Mary in intercession and distribution of graces to an entire sinful and disobedient group (mankind)? If one thing can occur, so can the other (so one might make a biblical argument from analogy).
In fact, the Bible frequently refers to what might be called “human distribution of divine grace” up to and including salvation itself:
1 Corinthians 9:22 (RSV) I have become all things to all men, that I might by all means save some.
2 Corinthians 4:15 For it [his many sufferings: 4:8-12, 17] is all for your sake, so that as grace extends to more and more people it may increase thanksgiving, to the glory of God.
Ephesians 3:2 assuming that you have heard of the stewardship of God's grace that was given to me for you . . .
1 Timothy 4:16 Take heed to yourself and to your teaching: hold to that, for by so doing you will save both yourself and your hearers. (cf. 1 Cor 7:16; James 5:20; 1 Pet 3:1)
1 Peter 4:10 As each has received a gift, employ it for one another, as good stewards of God's varied grace.
All of this being the case, it is nothing so unbelievable or extraordinary to believe in faith (in line with Catholic tradition) that God chose to involve Mary in intercession and the distribution of graces, even should He decide to do so in every case. God can do whatever He wants!
It is written in the Psalms and prophets that God could raise up a rock or a tree to sing His praises, if stubborn men refuse to do so. God once used a donkey to speak and express His will to Balaam. He can use babies, or infants, and the most “unlikely,” unexpected human beings. He appeared in a burning bush and in a cloud. He chose to come to earth as a baby!
Why should anything He does or chooses to do surprise us, or make us wonder in befuddlement? The ending of Job makes this clear enough. His thoughts are as far above ours as the stars are above the earth (Isaiah 55:8-9).
We were all meant to be sinless and immaculate. Now we are to believe that God couldn't or wouldn't raise one solitary human being to a sinless state before they get to heaven (where sin has no entrance), or to help redeem the human race (as a non-necessary “co-worker” of God, chosen by Him)?
Mary, being immaculately sinless and utterly unique as the Theotokos, can obtain far more than any of us from God, on the basis of the Scripture, “The prayer of a righteous man has great power in its effects” (James 5:16). We participate secondarily in our own salvation by consent and obedience; Mary intercedes for the salvation of all and helps distribute God's graces for that purpose.
God is the Redeemer. Mary is simply a helper or chosen vessel, just as Moses or John the Baptist or Elijah or Paul or Peter or John or anyone else was. In no way does this impinge upon God's sole prerogatives because He is simply using one of His creatures for His divine purposes.
Just as we are allowed the unfathomable privilege of participating in our own redemption, likewise God willed that the Blessed Virgin Mary, the Theotokos, the Immaculate one, the perpetual Virgin, the Second Eve, would play a part in the redemption of all. She doesn't (solely and sufficiently) cause the redemption any more than we (solely and sufficiently) cause our own redemption.
This was God's marvelous plan – to involve a creature and a woman at every step of the way. Eve brought down the human race, acting with Adam; Mary helped to raise it, acting in concert with Jesus Christ, her Son, the second Adam (as Paul describes Him).
There is nothing intrinsically impossible, excessive, idolatrous, or unbiblical in these beliefs: held by Catholics through the centuries and firmly entrenched in Sacred Tradition. It is not an a priori impossible or implausible belief to hold, from a biblical perspective. It's completely harmonious with Scripture.