There are some seriously world class teams (notable Einstein appearances) who work only off a few hours a week (like the 10-15 hours range). I can't think of them right now, but I know there's a CD post about them somewhere. My own personal team meets about 25 hours per week, but a lot of that time is non productive. I think if you come into every meeting with a really solid plan and a solid schedule for what you want and need to do, delegate correctly and don't get distracted, I think a lesser schedule can work well.
A lot of this next part depends on what you head mentor's positions are. For us, we have one mentor who comes in three times a week during our "mentor meetings", then one who is the faculty sponsor who is there for school reasons. If yours head mentors are complaining that they don't want to be there every day, mentor specific meetings are a great solution. Mentors come in for two two hour meetings per week then eight hours on Saturday. The mentors still have plenty of influence over what we do, and we'll only go through about 5 hours of work without having a mentor there. Every major decision is done at these meetings.
If your head mentors are teacher sponsors, there's another issue. Because of anonymity, I have no idea if you're a community club, school affiliated or what organization you are in. In this case, try and get more than one faculty sponsor. In your "working harder not smarter", you can figure out which sub teams (or whatever you call them) need to be there at each meeting. Everyone needs to be there for kickoff, but week two? Electrical doesn't need to be there (for most teams, some teams have their entire robot built week two

).
In dealing with your leadership team, in regards to their comment about "Screw them, they don't know anything about building robots", they are SO very wrong. I don't care who your mentors are and what they do, they are WRONG. Just about any mentor is at least going to have the experience purely on the team of just being on it for longer. Most likely, they also do some sort of STEM job, adding to their experience even if they don't specifically use robotics. Even if they are the most joe-schmo on the street (which I'm not saying they are, all of our mentors are wonderful and brilliant), they have more reasoning skills. Sometimes it's better to have someone who has a fully developed brain (human brains don't develop fully until age 25) to look at something and point out something.
Also, don't try and operate like a "real world company". Sure, real engineering is great, but you have virtually no experience here. You have to build in training and strategy into a even shorter area where an engineering firm doesn't. You shouldn't work like a company, you should work like a FIRST team. There is nothing out there you can compare this to. That's part of the charm. We've built up our own little community with our own standards, our own expectations and our own opinions on everything. You must create just about everything from scratch, so test it out. It seems like previous years haven't turned out well for you in terms of robot performance, so maybe this is the year you change things up.