Yes, the classical formula for friction does not involve surface area: F = mu, where u is the coefficient of friction. However this simple model developed by Coloumb, while accurate for classroom experiments like in that youtube video, does not factor in all the variables involved in traction in the real world. So saying wheel width has no effect on grip, or more specifically traction, may often be inaccurate.
You're not considering load sensitivity, sidewall flexing, slip angles at speed amongst other factors. It's similar to why wider tires are preferred for increased traction (especially on acceleration) for bikes and cars and stuff. Rubber and urethane, despite being different compounds, are both elastic and display many of the similar properties.
TLDR: Although friction is unrelated to surface area in controlled environments with smooth surfaces, wider wheels with larger contact patches still generally have increased traction if you control for all other variables.