Christ gave us to her care at His dying moment on the cross. However, that particular statement you object to was a private revelation by pope Leo XIII, you say, so it would not be in the scripture. Why don't you do what I asked you already and read why did he say what he said?
Uhhhh. NO.
Christ gave her care to John.
Some might even quite reasonably that He did so in a fairly strongly END OF AN ERA; closing off of an earthly relationship sort of way.
. . . a closing off of the MOTHER/SON relationship.
It seems to me, that her brief was as all of ours post Cross and REsurrection:
She was from thenceforth to see Him as Isaiah . . .
HIGH AND LIFTED UP; HIS TRAIN FILLED THE TEMPLE; WOE AM I A WOMAN OF UNCLEAN LIPS.