Yes, context is my friend.
John 6: 67-71
After this many of his disciples went back; and walked no more with him. [68] Then Jesus said to the twelve: Will you also go away? [69] And Simon Peter answered him: Lord, to whom shall we go? thou hast the words of eternal life. [70] And we have believed and have known, that thou art the Christ, the Son of God.
[71] Jesus answered them: Have not I chosen you twelve; and one of you is a devil? [72] Now he meant Judas Iscariot, the son of Simon: for this same was about to betray him, whereas he was one of the twelve.
Here there is a clear distinction between Jesus’ disciples, many of whom left Him, and the chosen 12, whom I believe He is referring to in the immediate next verses as His brethren. We already know that Jesus Himself uses brethren at other times when He is clearly not referring to relatives at all.
In any event, why would the brothers not believing in Him yet mean someone else needed to take care of Mary? I don’t see these two events as related at all. Not one brother was willing to stay with Mary?
Also, Jesus must have known that they would soon come to believe in Him. Why would Jesus use His dying breath to have Mary taken care of temporarily when she would have been fine in the short term in the care of other relatives or friends? We see in the Lazarus story (today’s Gospel by the way) how communities rallied around those who lost a loved one.
And what about the sisters? Since you translate brethren to mean brothers, do you mean the sisters also did not believe in Him? Jesus was 30, there must have been some sisters of marrying age, wouldn’t the sons-in-law have taken care of Mary?
Wouldn’t denying Jesus have actually made them safer from persecution, as it did for Peter initially, so they would have been better able to take care of Mary?
Love,
O2
One last possible point to ponder: Jesus was so fond of John and John's devotion that what Jesus did would have been such a tremendous blessing placed upon John: if there were no step siblings, the blessing is full; if there were step siblings who did not even come to His last moments alive, tragically confirming in their reasoning their worst fears about His preaching against Rome and the Jewish power structure, the fact that John was there and perhaps the youngest of His disciples, being given such a weighty responsibility would have been an enormous affirmation of Jesus's love and trust in John.
66As a result of this many of His disciples withdrew and were not walking with Him anymore. 67So Jesus said to the twelve, “You do not want to go away also, do you?” 68Simon Peter answered Him, “Lord, to whom shall we go? You have words of eternal life. 69“We have believed and have come to know that You are the Holy One of God.” 70Jesus answered them, “Did I Myself not choose you, the twelve, and yet one of you is a devil?” 71Now He meant Judas the son of Simon Iscariot, for he, one of the twelve, was going to betray Him.
1After these things Jesus was walking in Galilee, for He was unwilling to walk in Judea because the Jews were seeking to kill Him. 2Now the feast of the Jews, the Feast of Booths, was near. 3Therefore His brothers said to Him, “Leave here and go into Judea, so that Your disciples also may see Your works which You are doing. 4“For no one does anything in secret when he himself seeks to be known publicly. If You do these things, show Yourself to the world.” 5For not even His brothers were believing in Him.
John 6:66-7:5
These brothers are not his disciples. 7:5 notes they, the brothers, were not believing in Him whereas 6:69 indicates the disciples "have believed and come to know You are the Holy One of God."
John is clear in his terminology in these passages.