However, when I have visited churches that have such a mindset, sometimes this emphasis on being "one of the people" can go unintentionally awry: In some instances the middle-aged minister will be wearing clothing so contemporary and "hip" that they appear as genuine as my trying to use the word "hip"...or the phrase "with-it," like a father trying to use his teenage son's lingo. In other instances, the minister's attempt to "dress down," could almost be seen as disrespectful (t-shirts, floral short sleeves, etc). Perhaps the worst variation on this theme is the minister who wants to "not dress like a minister," but wears clothing so expensive and ostentatious that you're left wondering if the church may be paying the man too much, or if he realizes what an affront such clothing could be to the poorer members of the congregation.
You'll notice that the the pictures I have used to illustrate this entry come from what is now seen as the "evangelical" tradition of the Anglican spectrum (what was, for several hundred years, the basic vesture of Anglican priests)--the classic cassock, surplice, and tippet still worn as the standard by the Free Church of England clergy. In parishes and cathedrals in the United States and the United Kingdom that follow Percy Dearmer's scholarly admonitions, the surplice and tippet are worn for Morning and Evening Prayer, while an alb, stole, and chasuble are worn for the Holy Communion; in some parishes, the surplice and tippet are still retained for Communion, whereas in others a stole is worn over the surplice--in still other places the cope may be worn over the surplice and stole or tippet.
While there is variety in such practices, it is still quite clear that the priest is, well, the priest. You could swap out Vicar Smith for Father James or Dr. Williams and you could still tell who was the ministerial presence leading the divine worship of the Church, for he would be wearing a uniform clearly recognizable as that of an ordained minister of the Church. The priest doesn't represent himself, he represents the office the Church, and the Church as an extension of Christ's presence--and he holds this as a temporary cell in the Body of Christ: When he leaves, retires, or dies, another will take his place to perform the same sacramental and ministerial functions, and what this new priest does and how they are known is not a place primarily for the expression of individualism--the Eucharist should remain substantially the same (which is why the use of the historic rites is also paramount), the Baptismal rite should remain the same, the office of the priest carrying out these functions should display continuity with what has come before, not just recently, but reaching back in time to previous ages.
When the priest wears the cassock and the surplice, he identifies himself with the office of the priesthood to carry out his Christ-like sacramental care of God's people, to demonstrate God's presence in the world. The uniform of the priest constrains the priest in a way similar to the liturgy constraining the priest: What the priest preaches from the pulpit must (or should) conform to the doctrine of the Prayer Book, for the language of the Prayer Book verbally exemplifies the continuity of the Church. Similarly, if he dons the cope or the chasuble for the celebration of the Holy Eucharist (as they both descend from the same garment), he is identifying with the continuity of the ancient Church in the celebration of this Sacrament. What he does at the Holy Table is the same as what Lancelot Andrewes did, what Saint Augustine of Canterbury did, what Saint David did. When the people look to the priest, they should see not only themselves as humans (and the faults, frailties, and failings of the individual minister), but the work of Christ in His world through His Church. This is why, whether one is an "evangelical" or "Anglo-Catholic," or an "evangelical Anglo-Catholic," one should be identifiable as a priest of Christ's historic Catholic Church, so that when the faithful see him, they see not only him, but a local expression of the universal Body of Christ.