No, when Christ washes the disciples feet and tells them to do the same to others he is not saying that the greatest thing a person can do is physical service for someone else. Taking care of other’s needs is a great part of the gospel. And it is the domain of the Aaronic priesthood.
But the greatest work to be done in the church by the priesthood is Melchizedek in nature, and Christ is washing his disciple’s feet as an ordinance, and instructing them to give that saving ordinance to others.
That which is Melchizedek is greater than that which is Aaronic, and the greatest work the apostles could do was not to give temporal aid or offer themselves doing menial service. It was to go and do the work of the Melchizedek priesthood, bringing men to the saving ordinances of the gospel.
In fact, Joseph Smith taught plain as day that the twelve were not to waste there time doing menial service, for their work was the work of Melchizedek.
“The Twelve also are not to serve tables, but to bear the keys of the Kingdom to all nations, and unlock the door of the Gospel to them, and call upon the Seventies to follow after them, and assist them.” (TPJS page 109)