While an ancient tradition, the Chrism Eucharist as presently known is a surprisingly recent addition to the ceremonies of Holy Week, arising from reforms of Pius XII in 1955 and innovations of Paul VI in 1970. In general, Anglican versions of it have unfortunately been based on these recent Roman rites, which reflect a quite specific understanding of priesthood and episcopate, not ancient or catholic practice.
In the third and fourth centuries, by when oils had come to be widely used for anointing catechumens and the newly-baptized as well as the sick, they were all probably blessed according to need, in some cases by presbyters as well as by bishops. Oil of the sick, at least, could be also used by lay people from day to day at their own initiative, and Pope Innocent I and the Venerable Bede are among defenders of that tradition against the eventually-victorious clericalizing tendency more familiar today. Nonetheless the blessed oils are for the whole Church, not for clerics per se.
The idea of a Mass to celebrate the ministries associated with the oils emerged by the sixth century. While Medieval Roman liturgical books provided for the blessing of oils at the second of three Masses of Holy Thursday, in practice these blessings were often rolled into the rather full celebration of the institution of the Eucharist, the Mandatum.
The separate Chrism Eucharist was reintroduced in the Roman Catholic use in 1955, as part of reforms of Holy Week. Only in 1970 however did that Church add the renewal of ordination vows, which they and many Anglicans now treat as more fundamental to the rite than the blessing of oils. The readings provided in the Roman Rite and copied by many Anglicans (Isa 61:1-13; Rev 1:5-8; Luke 4:18-19) seem to be about ordination, not the ministries or sacraments associated with anointing.
The reason for this has much to do with inner-Roman Catholic politics; in the 1960s there were outbreaks of clerical disobedience especially in Europe, and the idea of renewing vows of loyalty to the bishop was taken up to address them. Apart from the fact that priests attended the Chrism Mass to receive the oils anyway, and were hence available for this ceremony, the connection has been built on the problematic idea that, as Pope John Paul II put in his Holy Thursday letter of 1997, clerical participants were gathering "round your bishop to commemorate with joy the institution of the priesthood in the Church." That is, Maundy or Holy Thursday is seen as much or more the founding of the ministerial priesthood as of the Eucharist. Some may find this astounding. I will merely say this is not a view all Anglicans will entertain.
It is doubly unfortunate then that the real purpose of the rite is often obscured, and what was originally a celebration for the whole Church with the bishop, not for the clergy alone or specifically, has become a sort of clerical love-in.
Anglicans could do this better, by going back to the roots of the liturgy and the meaning of the oils used for catechumens, for the sick, and at baptism. Some dioceses have extended the renewal of vows at the Chrism Eucharist to include deacons and lay ministers. This is attractive, but pre-empts the renewal of baptismal vows at the Easter vigil, which is the more important exercise (even for priests!).
More defensible might be a focus on specific ministries, lay and ordained, associated with the oils, and hence interpreting the newer aspect of the celebration through the older. The bishop might invite hospital chaplains and visitors, catechists and educators, to participate in this communal preparation of the oils which symbolize the formation and the healing of the Christian Church as a whole.
It might be too ambitious to suggest that the renewal of ordination vows be removed, but if it is included then this element should at least be handled more carefully as part of the whole Church's ministry and mission.