First off, this is a fantastic question. The following response is not very structured, rather, a quick journal of my thoughts on the topic.
In my experience, I’ve always found that as a student, the mentors (even those who I’ve had outside of FRC) that have benefitted the students most are the ones who make the mentor-student relationship a partnership. As with everything, there is a spectrum, and usually having a balance somewhere in between is good. My personal philosophy leans closer to being a family, however.
Mentors shouldn’t be secretive, except for discussions that relate to inter-student disputes or legal issues, since it’s generally accepted that students aren’t mature enough to handle problems of that nature, and that’s okay. As a general principle, mentors should make their best effort to work 1 on 1 with every student, get to know them, and be a resource for the student to come to for general advice or expertise. This may mean that mentors do work on the robot, and that is also okay! Mentors are the role models and set the attitude for the student body, so mentors should be willing to apply themselves, and encourage students to do so as well.
Students who are more comfortable with their mentors are more likely to voice concerns, ask for help, and apply themselves. One may make the argument that professional relationships are what students are being prepared for since that will be their future, yet conversely, successful workplaces worldwide are becoming less formal, and in the end, our program exists to inspire students to love the STEM disciplines, so why should mentors hamper that experience and push students away by being “professional”? At some extent, IQ doesn’t matter, EQ does - and when your role as a mentor is building exceptional people who demonstrate mature social qualities, emotional intelligence matters.
With all this said, it’s important to find a balance. It’s never okay to sacrifice student safety or privacy for the sake of being a “family”, nor is it okay to let discussions go off topic and lose productivity. This balance and culture is always set by the lead mentor(s) and quite simply, those who have high EQ and IQ, respect their students and peers, and enable people to be their best will be the most successful. Culture is set from the top.
As always, there will be good points for either argument - family or professional. But speaking solely from my experience, building a family is almost always beneficial to the student experience and for their well-being.