[Ed Law]

There are many excellent suggestions already. I do want to add my perspective on this to may be help you and others in similar situation.
The chaos you are describing points to the failure of the team. Nobody should point fingers to who is at fault because it is the team's fault for allowing this to happen in the first place.
An FRC team is just like any organization. It must have a business plan that spells out the vision and mission, goals, organization structure, job description and responsibilities, contingency plans etc. This is the only way that the whole team will move in the same direction instead of in-fighting. There should also be no us versus them, students versus mentors etc. You are all part of the team and working side by side.
The next important thing is to have one effective leader, a CEO / coach or whatever name you want to use. This leader's job is to follow the business plan and take the team to systematically achieve the team's short and long term goals. There should be no personal agenda. A good leader will listen to all ideas, and make a timely decision to move the team forward. A team cannot afford to have too many cooks in the kitchen and allow any adult mentors or parents or student leader to change the course. The leader needs to make the final decision. A good leader also knows he/she cannot be the bottleneck so some delegations will be healthy and necessary.
Besides one leader, there are many ways to organize a team. Every team is different. Learn from other successful teams and find a structure that works best for you. It should be clear to everybody that the leader/coach and mentors are there to help the students and the team to succeed.
You have also expressed concern about selecting the right student leader. I will share with you how my old team does this. At the end of the season, graduating senior team captains propose a list of possible candidates that they feel is best for the team. Rising juniors and seniors indicate their interest to be a team captain to the team leader. We use a decision matrix approach. We list all the criteria/qualities. The importance of these qualities may change from year to year depending on what the team leader feel the team’s need is the following year. Then the team leader will consult with mentors who have worked directly with the candidates and also consult the current team captains to get their input. All candidates will be ranked on each of the qualities. Then you multiply the score with the importance and get the sum. This will usually narrow down to only a handful of candidates. We also take into consideration on balancing skills and maintaining gender ratio. It has worked well for us. It is a process we follow to be as objective as possible. This is an example of a process that can be documented in the team handbook so it will be followed in the future. This also prevents a new person coming in and completely changes how things are done.
You have also expressed disappointment of a previous year robot. Somebody else already addressed this. Design is also a process. If the whole team was part of that process and had regular review of the progress, then the robot is your team’s robot so not one person or group should be blamed for its outcome.
I can go on and on. PM me if you would like me to give any other suggestions.