On the contrary, the towel and basin were there for feet to be washed by themselves (Genesis 18:4 & 19:2, Judges 19:21, 1st Samuel 25:41 and others). In that culture (and as an externality of wearing sandals), it was custom to have the feet of people who entered one's home washed. It had nothing to do with Passover specifically, but rather out of both respect and general desire for cleanliness. For example, look at the other known foot washing incident:
Luke 7:37 And behold, a woman of the city, who was a sinner, when she
learned that he was at table in the Pharisee's house, brought an
alabaster flask of ointment, 38 and standing behind him at his feet,
weeping, she began to wet his feet with her tears, and wiped them with
the hair of her head, and kissed his feet, and anointed them with the
ointment.
then later
Luke 7:44 Then turning toward the woman he said to Simon, "Do you see
this woman? I entered your house, you gave me no water for my
feet, but she has wet my feet with her tears and wiped them with her
hair.
Emphasis mine. It was a custom to have the things around for anyone entering the room - that a person entering the home would wash their own feet, or perhaps a servant would wash it for them. In addition, in Luke's account of passage at hand, you'll notice a different reason for Jesus getting his wash on, emphasis mine:
Luke 22:24 A dispute also arose among them, which of them was to be
regarded as the greatest. 25 And he said to them, "The kings of the
Gentiles exercise lordship over them; and those in authority over them
are called benefactors. 26 But not so with you; rather let the
greatest among you become as the youngest, and the leader as one who
serves. 27 For which is the greater, one who sits at table, or one who
serves? Is it not the one who sits at table? But I am among you as one
who serves.
While the disciples jockey for whomever is the best, Jesus lowers and humbles himself, stripping down like a servant to wash those same disciple's feet. John records this dialogue as well, emphasis mine:
John 13:12 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. 16 Truly, truly, I say to you,
a servant is not greater than his master; nor is he who is sent
greater than he who sent him.
There was no tradition here, until this point. This actually starts a Christian tradition, a sense of inward and outward humility. Paul's letter 1st Timothy 5:10 included "washed the feet of the saints" in instructions for the selection of widows who wish to be provided by the church. Paul also writes in Philippians 2:7 that Jesus took "the form of a servant, being born in the likeness of men." Peter objected to this because he didn't understand what Jesus was trying to do (Jesus says this to his face in John 13:7), which is totally within his character.
(all references in RSVCE)
