It is important to note that this Omega scenario presumes different things than the typical one used in the Newcomb’s Problem. In the Newcomb's case, we assume that Omega is able to predict one's thought processes and decisions. That's a very strong and unusual situation, but I can imagine it. However, in the A,B-Game we assume that a specific gene makes people presented with two options choose the worse one -- please note that I have not mentioned Omega in this sentence yet! So the claim is not that Omega is able to predict something, but that the gene can determine something, even in absence of the Omega. It's no longer about Omega's superior human-predicting powers; the Omega is there merely to explain the powers of the gene.

I agree with your objection, but I think there is a way to fix this problem into a properly Newcomb-like form:
Genes GA and GB control your lifespan, but they don't directly affect your decision making processes, and cause no other observable effect.
Omega predicts in advance, maybe even before you are born, whether you will press the A button or the B button, and modifies you DNA to include gene GA if he predicts that you will press A, or GB if he predicts that you will press B.
This creates a positive correlation between GA and A, and between GB and B, without any direct causal relation.
In this scenario, evidential decision theory generally ( * ) chooses A.

The same fix can be applied to the Solomon's problem, Newcomb's soda, and similar problems. In all these cases, EDT chooses the "one-box" option, which is the correct answer by the same reasoning which can be used to show that one-boxing is the correct choice ( * ) for the standard Newcomb's problem.

I think that allowing the hidden variable to directly affect the agent's decision making process, as the original version of the problems above do, makes them ill-specified:
If the agent uses EDT, then the only way a hidden variable can affect their decision making process is by affecting their preferences, but then the "tickle defence" becomes valid: since the agent can observe their preferences, this screens off the hidden variable from any evidence gathered by performing an action, therefore EDT chooses the "two-box" option.
If the hidden variable, other other hand, affects whether the agent uses EDT or some corrupted version of it, then, the question "what should EDT do?" becomes potentially ill-posed: EDT presumes unbounded rationality, does that mean that EDT knows that it is EDT, or is it allowed to have uncertainty about it? I don't know the answer, but I smell a self-referential paradox lurking there.

( *
One-boxing is trivially the optimal solution to Newcomb's problem if Omega has perfect prediction accuracy.
However, if Omega's probability of error is epsilon > 0, even for a very small epsilon, telling the optimal solution isn't so trivial: it depends on the stochastic independence assumptions you make. )