What relevance did Christ's status as a condemned criminal have in comparison to His status as Lord, in terms of the authority He exercised in tasking John with Mary's care?
Of course, +John made no claims. Jesus informed him of what his free will would be. :) That's what the scripture says.
Where does Jewish Law say that the other children would have "made a claim" AGAINST a widowed mother who was already provided for? (And what would Roman law have to do with this?) Note that your scenario presumes that the children would have fought the decree of Jesus. Why would they have done that, Jesus was the oldest?
What do you mean by "witnesses"? While Mary was of course not male, she was the matriarch, and she was the only believer. I think her own testimony in front of her children, who all RAN (or never showed up at all), would have counted for something. It was only HER life. :) Mary was at the cross and suffered greatly. Would you have questioned her witness as to what our Lord said while nailed and dying on a cross, while you were hiding? I don't think they would have questioned their mother either. :)
Her Child was dying for all mankind on the Cross. There is no mention in the Bible of her having any other children. Church teaching has always held that she had no other children. The preeminent Protestant Reformers all agreed that she had no other children. Brother of Jesus DOES NOT mean son of Mary, if you believe otherwise that is your prerogative, but it doesn't mean you are correct.