“And when thou hast sacrificed [the ram], thou shalt take of his blood, and put it upon the tip of the right ear of Aaron and of his sons, and upon the thumbs and great toes of their right hand and foot; and thou shalt pour the blood upon the altar round about.” (Exodus 29:20.)
So yeppers, you got your priests with feet specially washed by the High Priest at the Temple, so that they’d be nice and ritually clean to have sacrificial blood put on ‘em as part of the Covenant. So Jesus wasn’t being host-y, He was being High Priest-y.
And that means that anybody who sits down to get their feet washed by a priest is taking on a pretty heavy priestly responsibility, at least symbolically. I’m not sure why this additional verse isn’t added as an explanation, unless it just grosses people out. In which case, people should wake up and smell the sacrificial blood of Temple Judaism. Seriously, the whole Jewish people got splatted with blood sometimes, so it’s a little late for us to be grossed out.
And there’s a nice direct connection with Catholic priests traditionally having a special care for the right thumb and forefinger that will be handling the Host.
Now, to be fair, someone who was having a “guilt offering” or “trespass offering” offered for him would also get smeared with the blood of the Lamb in the same places, so there’s some connection to Baptism also. (Leviticus 14:14) But in that case, it makes footwashing much more of an “I’m a sinner and you’ve got to fix me up” moment than a mutual service moment. I suppose you could argue that Jesus was instructing all the heirs of the Apostles to go to Confession with each other, but that’s not the way tradition has interpreted it. I suppose you could also interpret it as that Christians are supposed to die for one another. But yeah, not a harmless happy mutual service gesture.
Jesus was preparing the Apostles for His Sacrifice as the Lamb of God, so that they could act as His under-priests and help offer Him up to the Father, just as Aaron’s sons and all the regular Temple priests did. Likewise, he was preparing them to have their sins, past and upcoming in the next few hours, be forgiven by His Blood. He was also setting up both them and the heirs of their office as priests forever in the order of Melchizadek, to offer God’s Paschal Lamb continuously until the end of the world, and to be commissioned in a few days with the power to forgive sins.