The problem genuinely does have to do with the use of an impossible probability distribution. It’s not just a straightforward mistake (as in the wrong answer to the Monty Hall problem, say).
Now, you have no idea which it is. And the fact that you're holding a $20 doesn't tell you anything.
The $20 is fake information. It hasn't revealed any real info, as you knew you'd open something. The key is not to combine a world where the total envelope value is $30 and where the total envelope value is $60 which is what happens if you add up probabilities.
Instead, there's a 50% chance you're holding 1/3 of the total value, 50% chance you are holding 2/3. So if you switch, half the time, you add 1/3, half the time, you lose 1/3. net expected gain from switching: 0.
You have a pair of random variables (X, Y) that take values (100, 200) or (200, 100) with equal probability. Then E[X] = E[Y] = 150, there is no question about that. Also, E[X/Y] = 1.25, there is also no question. The only question is why E[X] is useful and E[X/Y] is not useful for our decision making - and I honestly don't know why.
The problem states that (and is only interesting because) one of the envelopes contains twice as much as the other, but not how much, so that the amount in the first envelope tells you nothing.
This is actually impossible because there is no information about how the amount (let's say the bigger of the two) is distributed, which usually implies uniform distribution. But a uniform distribution is only possible if you assume an upper limit (otherwise, what's the expected value?). If there is an upper limit, the question again becomes quite easy (you switch if the first envelope contains less than half the upper limit).
In this case, the argument that by switching you get 125% on average still holds!
I'd claim that this is even more interesting and paradoxical than the scenario you're talking about, where your explanation is correct (the distribution of the bigger value is crucial for your decision, and you know nothing about it).