You’re mixing up equilibrium payoff and Nash Equilibrium (which is a strategy profile). Yes, the payoffs of player A randomizing and player B playing scissors

You’re mixing up equilibrium payoff and Nash Equilibrium (which is a strategy profile). Yes, the payoffs of player A randomizing and player B playing scissors is the same as both players randomizing, but observe that this profile of strategies fails the ex-post facto definition of Nash Equlibrium, i.e. in retrospect player A would want to change their strategy choice to “play rock every time.”