In John’s gospel, the washing of feet is instituted by Jesus, with similar words used for the bread of life discourse. “if you don’t let me wash your feet, you have no life in you”, etc. For which Peter said, “then Lord, not only my feet but my hands and head as well”.
Why do we tend to down play this command of Jesus to wash each other’s feet? We rightly defend the Eucharist as a literal interpretation of John Ch 6 and treat as a sacrament. Some people do the washing during Holy Thursday service, but not the rest of the year, and most people have never had their feet washed by priests.
The washing of the feet occurred at the Last Supper. It’s action was symbolic.
Below is the passage you are speaking of. It is not at all the same as the Bread of Life Discourse. (See bolded line.) The Haydock Commentary has this to say about verses 8-10:
“The foulness of the feet, when the rest is clean, signifies the earthly affections, and remains of former sins remitted, which are to be cleansed by devout acts of charity and humility.”
The washing of the feet is an act of humility and without humility and seeking God’s forgiveness, we will have no part of Him. The washing of the feet also shows that the priesthood is one of service to the rest of the Church.
John 13:1-15
1] Now before the feast of the Passover, when Jesus knew that his hour had come to depart out of this world to the Father, having loved his own who were in the world, he loved them to the end.
2] And during supper, when the devil had already put it into the heart of Judas Iscariot, Simon’s son, to betray him,
3] Jesus, knowing that the Father had given all things into his hands, and that he had come from God and was going to God,
4] rose from supper, laid aside his garments, and girded himself with a towel.
5] Then he poured water into a basin, and began to wash the disciples’ feet, and to wipe them with the towel with which he was girded.
6] He came to Simon Peter; and Peter said to him, “Lord, do you wash my feet?”
7] Jesus answered him, “What I am doing you do not know now, but afterward you will understand.”
8] Peter said to him, “You shall never wash my feet.” **Jesus answered him, “If I do not wash you, you have no part in me.” **
9] Simon Peter said to him, “Lord, not my feet only but also my hands and my head!”
10] Jesus said to him, “He who has bathed does not need to wash, except for his feet, but he is clean all over; and you are clean, but not every one of you.”
11] For he knew who was to betray him; that was why he said, “You are not all clean.”
12] When he had washed their feet, and taken his garments, and resumed his place, he said to them, "Do you know what I have done to you?
13] You call me Teacher and Lord; and you are right, for so I am.
14] If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another’s feet.
15] For I have given you an example, that you also should do as I have done to you.