I feel like I'm in a wrong place, should I quit my PhD?
I am so sorry that graduate school is taking such a toll on your psyche. Please believe me when I say that you are not at all alone in feeling this way, and I agree with Buffy that this largely sounds like imposter syndrome -- something that even the top academics fall trap to.

In regards to your cohort, here is my advice, regardless of whether or not you change programs: remember that there will never be an even playing field. All of you have different backgrounds, strengths, areas that need growth, goals, and research interests. There is absolutely no way that any of your cohort members check every single box. With small cohorts like yours, it is difficult not to compare (especially if your program has set up a culture of competition). However, these comparisons can be detrimental to your psyche because you are only seeing part of the picture: what your cohort members publicly present. It could very well be that you are the strongest of your cohort in areas like grant writing, interacting with certain groups of people, or even a personality that others like to be around (a major factor that might sometimes get overlooked), but you are so early in your program that only time can reveal this. Wanting to succeed in academia is half the battle; if you have this, no matter how much or how little you are bringing with you into the program, you will find your fit. There is a reason you have been selected that might not be particularly clear to you, but it was clear to the graduate admissions committee, and it helps to shift your focus from what you think you lack, onto what your strengths are/what you offer that your cohort cannot.

In my own cohort, each of us offers something different. I highly respect the intellect of all of my colleagues, and when I publicly state this, some seem surprised and have even admitted that they had not recognized x as a strength of theirs until they heard it pointed out to them. Though I cannot guarantee you will get your a-ha moment, I can guarantee that early academia makes it very easy to forget the qualities you possess that have gotten you to where you are now. There have been occasions that your cohort members have lauded you or even felt jealousy, but since this knowledge is not explicitly expressed, you are understandably feeling as if you are not "enough."

Another thing to remember - your required classes are only one small part of your PhD. You will need to learn interpersonal, research, professional, administrative, and teaching skills, many of which can be learned but the extent to which you succeed cannot. These are skills you will need no matter the school that you choose, and it is often expected that you will accumulate them on your own (with some help from your program, however it is not necessarily your program's responsibility to walk you through them). It is so very likely that despite not coming in with the same knowledge or strengths as your colleagues, you will be stronger in certain aspects of the PhD track than others, and hopefully, you will be provided an adviser who can help you assess these strengths and weaknesses with you. For me, creating an Individual Development Plan (IDP) helped tremendously.

In regards to changing programs, are there programs that allow you to enter with an adviser? If you are not provided with this adviser, is it still possible for them to be on your committee? In my program (based in the U.S.), admission is heavily contingent on the adviser you choose and committee members are just as important. At your current institution, is it the norm to meet with professors you are interested in working with, to get a better sense of what they are about? Despite the program seeming to be restrictive on academics, perhaps there are faculty members that do not feel this way and can become allies to you. Find allies within the department, let them know the surmounting stress you feel, and gain their advice. Assess whether or not you feel comfortable talking with your graduate chair (if applicable). How accessible is the person you wanted to become your advisor, and how much time do they have for their students? How many students do they have? What are they even looking for in students - is it more based on research interests, or is it on grades? I believe that no matter the discipline, it is often the prior; since you are in a PhD program, you will be there a while, and your position in the department can change for better or worse at any moment and thus should not feel set in stone. 

If all else fails, know that people do change programs to find a place with a better "fit." Finding an institution that fits your needs might take time and visiting many different institutions you had not previously considered when applying. It sounds like you have learned a lot about yourself in this process, and if you are adamant in choosing a new program, there are people in your institution that will support you and help you with this; hopefully, in your department, but at the very least, within the graduate school, academic advising, career advising, and graduate success initiatives.

Your questions regarding how to approach the situation when reaching out to schools is not easy, but it sounds like the problems you are having can be voiced in a manner that speaks about department "fit," which the right school will understand. I am unsure how you chose your current program, but speaking in person or on the phone with prospective schools and advisors is really what solidified both mine and my adviser's choice to work together. Perhaps explaining your problems with your particular program to other program's graduate coordinators might put things into perspective and allow you to see how other programs stack up on the idea of PhD flexibility. 

On your qualms about speaking up affecting your funding - is your funding contingent upon your timely progress in the program? If so, keep focusing on that while sifting out your personal feelings about the program in the background. Focus on passing those exams. Take extra initiative to get to know your department, so they can get to know you personally and your POTENTIAL rather than your current knowledge. People in your department may talk, but the extent to which they have control over your ability to stay in the program funding-wise may be minimal, and letting them to get to know you outside of hearsay will be instrumental in making a good impression - a REAL impression of you. 

Changing programs or not, you got this. I cannot emphasize enough that programs care more about your potential than how well you are doing in these classes, because academics can be taught but your willingness to learn cannot. It sounds as if you are more than willing to learn and more than capable of showing your value to your program (or to a different program that might fit your needs better). Best of luck!