Caution: Long post. tl;dr, I agree with some things and disagree with others, and there’s some history at the end that you might find interesting.
You’re partially right. You may want to read the mission and vision statement again, but in general I’d say you’ve gotten the gist of it (at least as relates to this discussion).
The problem is that you are in the FIRST Robotics Competition. Therefore, there should be at least some playing to win. It’s a competition.
While I could go through and tear apart this statement for “why we should use tank tracks”… let’s go with it’s late, and I think a little thought will show what I might be getting at for COTS tank tracks. (Hint: “lack of” may be an operative word for COTS but not for custom–if you figure out why I say that, you’ll know what I’m thinking.)
Your test setup is OK, I think–it’s a fairly standard comparative friction setup–or you could leave the board flat and use a scale to pull/push the weight. Either method will work.
For extra credit, prove to a physics teacher that larger area of tread sample has more friction force than a smaller area. Same setup, slightly different methodology. (FIRSTers will tend to believe you anyway, particularly those who played Lunacy.)
Caution: Long history section. Read carefully if you opt to read it.
Here’s a thought on the overall use of tank treads: The trend for fields has been towards flat, fairly open (ish) fields. Tank treads as a drivetrain are generally speaking best/most needed over rougher terrain. I remember seeing quite a few back in 2003 (copper-mesh ramp with HDPE at the top), 2004 (HDPE platform, 6" step up), 2006 (steep diamond-plate ramp, PLUS using the extra traction as a stay-put). With the exception of 2016, no games in the last decade have had terrain tough enough for most teams to consider using treads. 2016 was hated by field-build groups because of the tough-to-build defenses–that sort of thing isn’t likely to make much of a return.
During that timeframe, and even back a few years before then, 6WD (and occasionally 8WD, a derivative) has been the absolute dominant drivetrain. It handles most “flat-field” terrain features (like the Depot last year) easily, can tolerate failures in such a way that half the drivetrain isn’t totally crippled, and is pretty simple for beginning teams (at least the KOP variant is–a true West Coast isn’t as simple but can be done with some effort). Before the 6WD was in wide use, the 4WD was king, at least for a few years, and around that time was when there were some interesting field obstacles where a lot of teams went for treads for various reasons (including that the KOP stock drive, when it existed, was, er… I can’t say the words on this forum–that stopped being an issue in 2005).
The thing is that there are a number of teams who have moved away from treads after mastering them. I pointed out FRC703 and FRC501 in an earlier post. 703 decided that they didn’t like treads, and proceeded to build a wheeled drivetrain that had nearly the contact area of the treads, and acted like a tread drive, but did not have treads! That’s right: 14 wheels, 7 per side, in an arc, gear-driven. I believe a later version had 10 wheels. Just to NOT use treads. Hmm… I wonder why they might have done that? And back in 2015, treads came up (for the scoring platforms) and 501 chimed into the discussion with this gem and was backed up by another team.