Why is it that some denominations follow the ordinance of humility as described in John 13 while others do not? What reason is there for not performing the foot washing ceremony before partaking in the Lord's Supper?
While I am not familiar with any denomination that practices footwashing as part of communion celebrations, I can address an issue that is implicit in the questions.
Descriptive is not the same a prescriptive
In other words, just because the Bible describes some people as doing something that doesn't mean all Christians should always do that too. As an extreme example the Bible tells us of bad things done, even by people described as 'righteous, such as the adultery of King David, and Peter's denial of Jesus. We are not expected to follow their examples. We have to use judgement, and follow Biblical guidelines, as to what is an example to be followed and what is description.
The Lord's Supper itself is not just descriptive. As well as describing what happens, Jesus' command is recorded: "Do this as often as you eat/drink it". No such command is attached to washing the disciples feet. In fact Jesus makes it pretty clear that what he is doing is not actually about feet, but about leadership and humility. Jesus expects leaders to act with humility towards their followers and be their servants. Actual literal footwashing may or may not be involved. Most Christians would say that a leader who carried out literally the act of washing people's feet, but behaved arrogantly towards the people they lead was not carrying out Jesus wishes.
The denominations descended from the "Schwarzenau Brethren" practice the feetwashing (as well as a common meal) in conjunction with the Communion. These groups would include The Church of the Brethren, The Brethren Church, the Grace Brethren, the Dunkard Brethren, and the Old German Baptist Brethren.
The Scripture is altogether clear in stating that Jesus washed the feet of His disciples, and that He gave them an example commanding them to do as He had done (John 13:14). The reference in 1 Timothy 5:10 is evidence that the early church kept up the practice which Jesus had earlier instituted.
The ceremony speaks of humility. The disciples had seen servants washing their master’s feet — but for the Creator of the universe to gird himself with a towel and get down on His knees in front of an illiterate fisherman and wash his feet – was unheard of. Our besetting sin is often an undue desire for status, and because each of us is inclined to feel he is above others, we need a service that will bring us on a common level.
The ceremony speaks of cleansing. John 13: 10 says, “He that is washed (bathed) need not save to wash his feet.” Two Greek words for “wash” are used here. The person who has been washed from his sins in the blood of Christ (as symbolized in water baptism) doesn’t need to be saved all over again when his feet become soiled as he walks through life. Our “feet” do become soiled; some of the filth of the world rubs off on every one of us; and thus we need repeated cleansings. The feetwashing service symbolizes the fact that we haven’t reached perfection; we still need cleansing.
The ceremony speaks of service. In a day when social service is being stressed so much, it is a happy thought that at least some believers observe an ordinance which symbolizes “service”. When we engage in feet-washing, we are promising to show in daily life what we practice during the ceremony in symbol. We are making a commitment to help our fellow brother clean up the mud when natural disaster strikes, nurse a wound when accident comes, and sit by a bedside when sickness invades the body.
Our Lord asks us only to do simple things. He set the example of feet-washing and tells us to do likewise (John 13:15).