1. That parents decide who their son's mohel will be
2. There are different methods employed when circumcising
3. There are hashkafic (worldview) differences employed by some religious camps in their preference for method
4. While there are halakhic differences associated with the worldviews, and while people may have a preference for how things ought to be done, with extremely rare exception, no one will say the brisses as described below are invalid. (When they ARE invalid, assuming the correct amount of foreskin has been removed, the method of fixing a bris is called "Hatafat Dam Brit")
5. The Standards that some rabbis and synagogues have employed in determining how they want a bris performed under their influence/oversight
If you are worried about the pain your baby will experience, you should be asking about what numbing options the mohel offers, how fast he works, whether he uses a clamp.
If you are concerned about sterility, you must ask whether the mohel wears gloves, how he sterilizes his instruments, whether he does metzitzah with his mouth, with a tube, or with a sponge
If you are concerned about methods and safety, you have to ask about clamps, freehand, whether the mohel marks the foreskin with a marker
Baby's grandparents, your rabbi, your friends are certainly allowed to make recommendations to you. But the decision of whom to hire (unless YOU CHOOSE TO DELEGATE IT) is your decision to make. The choice you make might come in conflict with the standards delineated below (in the STANDARDS section), but don't balk if you are confident in your decision that the mohel you've chosen is right for you and your baby.
Within those groups, there are different styles.
Clamp-using mohels (NOTE: I DO NOT USE A CLAMP) will most often use a Bronstein/Mogen Clamp, and very rarely use a Gomco Clamp. In all likelihood, someone who uses a clamp will also employ the use of a hemostat in order to achieve "Milah U'Priah B'Vas Achas."
Within the shield style, some will grab the foreskin with a hemostat, some will grab it with their fingers alone, sans-any other instrument.
The freehand mohel uses no instruments other than a knife, and does so because he thinks his method is least painful to the baby, and most reflective of the method utilized by mohels from over 1000 years ago.
And then there's metzitzah. Which is either accomplished through a sterile tube, through the mouth of the father (or mohel), or sidestepped completely.
Like any trade skill, a mohel develops a routine and system that works for him. When a mohel is asked to do a bris in the manner he is not comfortable doing, he can try to comply or he can say "I work best using the method I am used to. If you want me, this is the bris you get. You don't like that style, by all means, please call someone else."
When a rabbi is asked for a recommendation of a mohel, the rabbi (or his congregation) might have a set of standards for what they allow/don't allow. In my experience, some shuls use any or all of the following standards of operations:
A. Metzitzah may not be done with direct oral contact (out of concern for the transmission of any kind of disease), and may only be done with a sterile pipette (usually glass or one-time disposable sterile back of a syringe)
B. Do not allow a clamp (for Halakhic reasons)
C. Insist the mohel wear gloves.
D. Orthodox synagogues will likely insist on the mohel being an Orthodox Jew.
Hashkafically, some communities do not like the hemostat and they prefer metzitzah with direct-oral contact, but they do not call into question the validity of a bris performed with a hemostat (it is not viewed the same way that a clamp is viewed) or the metzitzah performed with a pipette. If they insist that the bris be performed without a hemostat (which most mohels use), they are essentially telling their constituents, "We have a very limited number of mohels we allow to perform a bris here. If these mohels work for you, great. If not, enjoy doing your bris elsewhere."
Most people will certainly agree that standards which protect the baby should be unquestioned.
This is why I think that the institutions/rabbis which insist there is ONLY ONE ACCEPTABLE METHOD OF PERFORMING BRIS MILAH do their constituents a disservice, as they are basically dictating to parents, "We don't trust your judgment about the mohel you've picked. You have to trust our judgment." Meaning, if the parents want a certain method, which happens to hashkafically differ from the rabbi's perspective (even while halakhically being extremely mainstream and acceptable), that IS OK. And the rabbi needs to be a little flexible. And a little more humble. Especially if he is not a mohel himself and does not know everything about Bris Milah. Preach all you want, but allow people to come to Judaism and decisions on their own terms.
Yes. There are some places where I feel it is important to draw a line (I think, for example, that the Gomco clamp is a torture device, and should never make an appearance at a bris). But there should also be a little bit of flexibility, especially when the bris is halakhically valid "lechatchila" according to virtually everyone, and the baby emerges with a fine circumcision and without danger of infection from the method employed on him.
In short, those who make standards for their institutions need to be very careful to be consistent, and must also be ready to explain to their constituents why certain mohels that everyone else hires might not be acceptable in their own synagogue.
There is an element of risk as well, because the family might opt to not do the bris in that facility altogether, and might be turned off from the institution which dictates to them how to live their lives.