But that's not how the story in Scripture goes. Mary noting the lack of wine (or being made aware of it) asked Jesus. He responded to her, then she told a third party, the servants (essentially, the kitchen crew) to do whatever Jesus told THEM to do.
And taking the story literally, if we pray to Mary, and she goes to Jesus, and he says “Woman, what have I to do with thee?”, it doesn’t sound like it’s very helpful, especially when she comes back to us and says “Just do whatever he says, he won’t listen to me”.
But that's not what it says literally in English. or Latin. or Greek. The members of the wedding party never asked Jesus directly, NOR did the servants ask what they should do. There's nothing about Mary saying "he won't listen to me" or anything similar.
But your description ignores the actual words of Jesus to his mother. They are not easy words, and it’s a lot easier to tell the story if you ignore what Jesus said to her.
Again, if you want to take the story to a deeper meaning than it appears to have, the servants are not like us, because the servants did not know Jesus, nor would they be “believers” in his power. So Mary would be the person who brought them to a knowlege of Jesus. But she did so as a human on earth, and her words was “do whatever he tells you”, not “I’ll ask Jesus, and I’ll tell you what to do”.
And after her human act of telling them about Jesus, she left them to communicate directly with him. Although as I say this isn’t a story of redemption, or even of a loving Father who answers the prayers of his children, it’s about Jesus manifesting his power to signify his authority and Godhood.
So we don’t know if they said anything to Jesus. We know that they came to Jesus, Jesus gave them a command, they followed the command, and Jesus performed a miracle.
There is nothing in that story to suggest that we, who already know Jesus, should have to be brought to him by Mary, or told to listen to what he tells us.
Further, there is no scriptural evidence that any of his disciples went to Mary to get access to Jesus, or that any other people who were already following Jesus around, or heard of him, ever went to Mary first to get an introduction.
Nor is there any indication in the Acts of the Apostles that Mary ever led a single other person to her Son, for any purpose.
In fact, as you point out, the story of Cana doesn’t even tell us if anybody went to Mary to ask for help or not. It could well be she noticed the wine was out, and went to Jesus on her own accord, rather than in response to supplication by others.
So I still fail to see how, even looking for a deep, hidden meaning in what appears to be a rather understandable story (with only the rebuke to Mary being hard to understand), how you could conclude that the story indicates we, who know Jesus, as believers, should ever ask Mary to go to Jesus for us.