The first area of discussion is rubrical, and has to do with the fact that the rubric expects that it will be men - male gender - who will be chosen to have their feet washed. As soon as the question is asked about the possibility of women being among those having their feet washed, a new agenda of male/female equality is imposed onto the rite, when the rite itself can make no sense of such an agenda. This is so even if the parish priest declines the request, and takes the subsequent flak. One priest's response to this is to choose this year to omit the rite of the washing of the feet .
The second area of discussion (which, in the realities of parish life, often occurs along with the first) is to see in the Mass of the Lord's Supper an indiscriminate celebration of ministries of all kinds in the Church. The Mass of the Lord's Supper then becomes the occasion for the renewal of commitment or calling of lay ministers of Holy Communion, readers, and so on. This question is raised in this discussion, particularly in my own comment, though the discussion focusses mainly on the role of extraordinary ministers of Holy Communion. This is all based on seeing the rite of the washing of the feet as a symbol of "service" in the Church, "service" being understood in a wide and all-embracing way. There are certainly times when it is appropriate to place the celebration of a blessing or induction for a particular ministry into a liturgical celebration such as Mass. But usually this will be a celebration of Mass on an occasion when the Mass does not have an exceptional character of its own. The Mass of the Lord's Supper does have a quite exceptional character of its own, which I think militates against such an addition of para-liturgical celebrations - such additions or insertions just "don't belong".
The key to getting this right seems to me to be the rubric in the Missal that refers to the homily during the Mass of the Lord's Supper:
The homily should explain the principal mysteries which are commemorated in this Mass; the institution of the Eucharist, the institution of the priesthood, and Christ's commandment to brotherly love.
1. As a celebration of the institution of the ordained priesthood, it is not appropriate to see in the Mass of the Lord's Supper a celebration of other (lay) ministries. It is an utter derogation from the meaning of this liturgical celebration properly understood. This removes the second area of discussion mentioned above.
2. The commandment to brotherly love needs to be seen as a celebration of Jesus's love shown towards the Twelve; the love of Jesus (its Head) towards the Church (his Body, represented in the Apostles); and the love of Jesus towards the Church that is continued today and expressed in the service of the priest towards his people. The integrity of this representation needs men to be chosen to have their feet washed, so that the service of Jesus to the figures of the Church is correctly represented. This removes the first area of discussion mentioned above.
To summarise, from memory, the homily that I heard yesterday evening.
Jesus loves those who were his own "to the end". He wants to give himself to them, definitively and for ever. So he hands himself over to them in his Body and Blood that are the Eucharist; and he hands himself over to them in the Church, represented by the Apostles. Jesus still loves us "to the end" in the Church. He needs priests, the Church needs priests, for this to occur in the Sacraments, and most especially in the Eucharist. This is the service of the priest to his parish - in celebrating the Sacraments and most especially the Eucharist. The washing of feet by the priest represents this service of the priest to the parish.
The sense of "Christ's commandment of brotherly love" referred to in the rubric on the homily is therefore profoundly related to the priesthood as the visible sign and existential realisation of that love. And this is the correct sense of the rite of the washing of feet.