The point is that if your strategy becomes a known quantity, it can be strictly dominated by your opponent choosing the strategy that wins against it. This would push you both back towards random choice (whether iterated or one-off).

A Nash equilibrium says that each player plays optimally given what all the other players do. If one player plays deterministically, it's no longer optimal for the other one to play randomly, so it's not an equilibrium.

Setup your own experiment and let me know what you find.