The only real thing about Mediatrix is that:
“No graces given to man are given without the intercession of our Lady.” (teaching level: pious and probable).
A secondary meaning is that our Lady freely gave her consent, and the Source of all grace was born into the world (teaching level: certain).
That’s really all it is.
I intensely dislike analogies like Mary “distributes” the graces as if she were some sort of treasurer: here’s one for you, one for you, two for you, four for you, none for you…
I dislike the “neck” imagery as well, for the same reasons as the “distributes” imagery. Reason given above: the Sacraments, which work ex opere operato. No “distribution” by Mary.
I also disagree, very strongly, with the idea that she is a mediator between man and Christ. The Incarnation itself tells us this cannot be, as only the God-Man can be Mediator in the strict sense. There is no need for Mary to mediate between Christ and man, because Christ himself is man and God. This is his very role as both Redeemer and Eternal High Priest, and no intermediary is required between man on earth and Christ. Adding this role approaches violation of Scripture (cf. 1 Tim 2:5; Heb 7:24-25) and the theology of the Sacraments.
No, Mary is an intercessor, and an intercessor par excellence. The concept of Mediatrix simply tells us that in the order of grace God has laid out from eternity, he has ordained that all graces given to man at least since the moment of the Assumption given (directly by God) are also due to the intercession for those graces by our Lady.
So when I was absolved of my sins directly by Christ (through the ministry of his priest), it was because our Lady prayed for me. When I received Communion and obtained sacramental grace, and Christ’s Presence itself, it was also because our Lady prayed that this would happen. When I was baptized and justified through Christ’s blood (directly), it was because Mary prayed that it would be so.
That is how I best understand (and I believe is the correct way) the title.
Oh, and for all this, it should remain at least sent. certa at the most. It should not be defined as dogma.