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Notes -
I think that this would not work. (Or technically, it would work in that it would make a maximally persuasive argument, but "maximally" still isn't "very".)
This falls under a class of arguments which goes "everyone who is wrong says similar-sounding things to everyone who is right, so you can't say that the other people are wrong and you're right". To use an example I used elsewhere, you can say "all humans are capable of enormous evil, if you were in Hitler's situation, you'd probably do evil too". It sounds plausible--a lot of people who don't do evil might do evil if they could get away with it--and it's impossible for me to disprove by observing that you're wrong about me.
But on the other hand, I am not in a position to do evil like Hitler, but I am in a position to do evil like Cade Metz. And I haven't.
The claim that I'd be like X in X's position isn't based on actually analyzing my personality, it's based on carefully setting up the claim so that it's hard to falsify, based on the fact that everyone will deny the claim whether it's true or not, and then doing a leap from that to "well, surely your denial means nothing".
I don't think such an oracle would work on me, for the same reason I don't think I'd be like Hitler in Hitler's position--I know myself better than you, and "everyone would deny this, so you are indistinguishable from the other denials" isn't actually good reasoning.
If an oracle would make the maximally honest, persuasive, verifiable argument to me that is either for or against homeopathy, and it picked pro-homeopathy, its argument would fail to convince me--for reasons related to there not being any good arguments for homeopathy. You may say all you want "a homeopathy believer could say the EXACT SAME THING about the other side", but it doesn't make me and him similar.
Ah, sorry, underspecified the thought experiment. The goal for the 'almost half' is to show that some people aren't persuaded or reseated in their positions. In the case of the oracle being asked about homeopathy and picking a pro-homeopathy side, it would be genuinely limited to incredibly weak arguments because no stronger honest ones exist, and a petitioner would only be confronted with a slightly clearly idea of what 'water memory' is supposed to be, and at most a slight 'isn't it weird' evidence that didn't have medical ramifications.
Similarly, the oracle can't persuade a homeopathy advocate who commits by having himself deafened before asking the question.
But those are a bit of a waste of a question. At best, you'd now be able to tell yourself you can out-think the oracle, or have slightly better arguments for a position you reasonably believe is already extremely strong. In these cases, you have no benefit to asking the question.
The thought experiment is more for questions in the middle, where you might have some confidence, but there are plausible arguments on each side. I don't, personally, like gun control, but there are sane arguments in its favor. If I asked this thought experiment's oracle, it could blast me with the most persuasive form of every verifiable argument for or against the policy. I might plausibly not comply, because there are underlying values assessments. Not optimistic -- I've changed my mind on this topic before -- but I'll recognize it as a possibility for the sake of the thought experiment.
The question is whether I should ask. The oracle is only providing information, and it's even information I can verify, so from that perspective it's totally beneficial. But it's also incredibly biased information, provided by something built to be as persuasive with that biased information as possible, and it demonstratably does change the beliefs of many people who speak to it and get a counterargument, or reduce the ability to doubt for many who it reinforces.
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