NavList:

The spring-based demonstrator is a good concept. I agree that the value of the instrument is educational and it should help cultivate sound intuitions.

Agree also that the "insensitivity near equilibrium" is itself important to appreciate and could help rationalize the level-of-effort and level-of-precision put into a final fix.  The group favoring mathematical exactitude can relax their tolerance bounds to domain-appropriate levels and do so not "just to be agreeable" but rather with a correct sense of the cost of imprecision and not an inflated rate.  The other group, thinking the mathematics is more trouble than it is worth,  has a better chance of feeling that the pull toward the short side of the triangle is natural and "or course true" while also being reassured that 'doing it right' is as easy as doing it wrong provided your sense of the marginal value of incrementally greater precision is well grounded.

You may be too quick to say that the device doesn't help explain the optimality of least squares (under set conditions).  You can hold the center-point cursor at the incenter of an arbitrary triangle, point out that the three springs are equally extended and so under balanced tension, then ask why the cursor moves when you let go.  How can two springs getting longer while one gets shorter possibly take you to a lower energy state?  I think you are in much better position at that point to explain the BLUE-ness of least squares.

I have thoughts on implementation that I may write up separately.  In my mind selecting the springs themselves is a more difficult problem than friction in the sliders. 

--Lyle Ruppert