Bill Cork has an excellent discussion of Eucharistic ministers and the nature of the Eucharist.
I served as an extraordinary Eucharistic minister for 6 years. Too long, in my opinion. I decided to resign last year because I could detect a sense of habit and routine setting in. This duty cannot be performed by habit or routine. I cannot take it for granted, so I needed to step back, even though there is a shortage of people to serve as EEM's. It's better not to do this duty than to do it poorly.
I look upon this duty as me simply serving as the extensions of the hands of the priest. He just didn't have enough hands to go around, and I am contributing my hands during a time of priestly shortage. Someday, the EEM will no longer be needed.
Even so, I was often overwhelmed by emotion after serving as an EEM. The faith of those who receive is radiant on their faces and my own poor faith is put to shame. Oh yes, there is the many casual recipients and many who I suspect do not have a clue or care about what they are doing. That is mostly not my call, but I do report any abuses, general lack of reverence or other problems to the priest.
I attempt to be "invisible" to the recipient. This is a meeting between the person and God. I just happen to be nearby. I try to be both reverent and unnoticed but will not hesitate to climb over pews to deliver the Body of Our Lord to an elderly or handicapped person who cannot go up to the altar.
The best aspect of EEM service for me was the time spent in our parish's Blessed Sacrament chapel. As many parishes do around here, the Tabernacle is not on the altar, but in a separate chapel. Ours is hushed, with soft indirect lighting. One light explodes around the back of the Tabernacle drawing you in. The tabernacle is plain, and there are about 10 simple kneelers in the chapel. One EEM is supposed to wait there whenever the tabernacle is open. When possible, I would take this tasks on, and would kneel untiuil the Tabernacle was closed after Communion.
Our Parish is named after a Franciscan saint who most famously served in South America
It is here that I find I can pray best and look deep into my heart. After Mass, a few reverent people come in kneel down and pray silently. I also usually retuen after Mass. We all are alone in our own ways and are careful not to disturb each other. The only sound I often hear is the sound of someone quietly crying. I add a prayer for that person's peace.
Often, when I hear that sound, I am the only one in the chapel.