The case where switching gives you 2A and the case where switching gives you A/2 describe 2 completely different universes, not two different actions in one.
Take the practical example of $100 and $200: A is either $100 or $200 depending on which envelope you receive first, so the equation is either 0.5*A + 0.5*2A, or it is 0.5*A + 0.5*(A/2). It can never be 0.5*2A + 0.5*(A/2)
The thing that helped me reconcile this was realizing that the A in the two equations are different values; then replacing them.
(0.5⋅(0.5⋅A + 0.5⋅2.A)) + (0.5⋅(0.5⋅A + 0.5⋅0.5⋅A)) = 9/8 A
Replacing the As with their actual values
0.5⋅(0.5⋅100+0.5⋅2⋅100)+0.5⋅(0.5⋅200+0.5⋅0.5⋅200) = 150
> If A is the smaller amount, then the other envelope contains 2A. > If A is the larger amount, then the other envelope contains A/2.
These are conditional probabilities so you can't simply add them up and compute an expected value of switching like they show in the example.