The word used by the Church does not have the connotation of something being "better" if it is extraordinary as it does in English. Extra-ordinary means that is not the ordinary, it is something other than the norm.
In the case of extra-ordinary Eucharistic ministers, they are not the norm and should only be used when necessary. The ordinary Eucharistic minsters are the bishop, priest, and deacon. However, if it is necessary (and unfortunately convenience is often confused with necessity), laymen who have been trained properly may also distribute Holy Communion, or take Viaticum to the sick.
It is similar with the Mass. Ordinarily the more recent form (the "Novus Ordo") is what you will find in most parishes of the Latin rite. However, there is also the older form of Mass (the "Tridentine") that may also be said. It is an extra-ordinary form, in that it is not what you will find in most parishes, however, out of necessity, ie people asking for the older form or priests wishing to say the older form (as long as it is not to the exclusion of the Novus Ordo), it can be done, and in the pastoral mind of the Church, it should be provided for those who wish to have that form of Mass.
When I say "to the exclusion of the Novus Ordo" what I mean is, that if there is a priest in a parish, and he prefers the "Tridentine," he cannot just change all the Masses to Tridentine if there are some faithful in the parish who desire the "Novus Ordo." This goes both ways as I'll point out below.
There seems to be a difference, but both in the case of the Eucharistic ministers and the Mass, the use of the extra-ordinary form is based on whether it is needed or not. If nobody asks for the Tridentine, then it is not needed to be done. However, as I stated before, if the faithful do request the older form or if priests wish to say the older form, then that is considered a necessity in the mind of the Church, and every effort should be made to accomodate that.
The pastoral mind of the Church goes both ways. If people don't want to be forced to go to a "Tridentine" Mass when they prefer the "Novus Ordo," then those who prefer the "Tridentine" should not be forced to go to "Novus Ordo" Masses.