That’s one take on it–and a very ‘egalitarian’ take it is, fitting in very well with contemporary social (or societal) thought, going all the way back to Lord Oliver Cromwell’s “plain man, warts and all” philosophy.
Not to be picky (I hope) but rather than thinking that “no one is better than the other” I believe the Church take is slightly different while not actually contradicting that above statement. While we are indeed all ‘equal’ as to soul, we are not at all equal otherwise. (Indeed, last Sunday’s gospel regarding the parts of the body proves just that. A foot does not equal a hand, but both are ‘parts of the body’. Paul even speaks of lesser parts needing greater protection!)
As for the ‘foot washing’ I believe this was done not to give the disciples a sense of ‘you are equals’ but instead, “you are servants”. That is a distinction which has very much gone into obscurity–the idea that we are to be servants of each other–in essence, to consider ourselves as low and others as greater. However, that distinction and idea is very prevalent not only in the gospels but in great Catholic works (St. Alphonsus Ligouri, Thomas A Kempis, St. Augustine, St. Thomas Aquinas, St. Teresa of Avila, St. Louis Grignon de Montfort, et. al.)
So I’d have to say that while I agree we are all ‘equal’ in our souls, I’m not in agreement that Christ through the foot washing in particular is specifically teaching us, through foot washing, that “we in the Church are all equal”. I do, for example, think that the saints were ‘better’ than say, I myself, and that’s why I look to them for guidance and help. I might never have the personal qualities of, say, the intellect of Aquinas (he will always be superior in that regard). But insofar as my limited intellect goes, I may be able to go as far in my ‘lesser’ ability to equal or even surpass St. Thomas’ ‘greater ability’. So while we are not ‘equal’ in ability, we are perhaps equal in faith or in desire to please God.
(As an aside–the ‘foot washing’ has become such a hot button topic in the U.S. that probably unfairly it has come to have meanings that were never part of it. Look at how many people now yammer to have “womens’ feet washed” --usually on the grounds of ‘equality’ to boot–no pun intended.)