Experts draw and shoot pretty darn quickly (link).
There's a "gun-slinger's paradox", but apparently the effect of that is not very significant.
If the assertion is true I'm going to guess it's because the person who draws first is relatively inexperienced/unpracticed/inexpert.
Specifically, an expert can **react** , draw, and shoot within 300 msec (see link above): so unless their opponent can **initiate** , draw and shoot in _less_ than 300 msec then they'll 'lose'.