I'm sorry, I didn't read the rest of your post after seeing the 0.5^99 estimate of the probability of being the original because the math looked very wrong to me, but I didn't know why.
While I agree there is nothing controversial about saying that after one step you have a 50% chance of being the original, I'm pretty sure it is not true that you only have a 25% chance after two steps.
Yes, if you are the original after step one, you have a 50% chance probability of still being the original after step two. So, I Oi is the probability of being the original after step i, then P(O2|O1) = 0.5, P(01) = 0.5, P(O2|~O1) = 0, so P(O2) = P(O2|O1) * P(O1).
BUT, after step 2, O1 is not 0.5, since there are 3 copies of you in existence, and 2 of those WERE the original after step 1, so P(O1) after step 2 is 2/3, so P(O2) = 2/6 = 1/3.
Don't think it affects the substance of anything else in your argument, though, but the math bothered me since intuitively the expected value of originals after any number of steps should be 1, which should be equal to the number of copies in existence times the number of copies.
If I'm missing something I'd welcome an explanation of what.

I'm sorry, I didn't read the rest of your post after seeing the 0.5^99 estimate of the probability of being the original because the math looked very wrong to me, but I didn't know why. While I agree there is nothing controversial about saying that after one step you have a 50% chance of being the original, I'm pretty sure it is not true that you only have a 25% chance after two steps. Yes, if you are the original after step one, you have a 50% chance probability of still being the original after step two. So, I Oi is the probability of being the original after step i, then P(O2|O1) = 0.5, P(01) = 0.5, P(O2|~O1) = 0, so P(O2) = P(O2|O1) * P(O1). BUT, after step 2, O1 is not 0.5, since there are 3 copies of you in existence, and 2 of those WERE the original after step 1, so P(O1) after step 2 is 2/3, so P(O2) = 2/6 = 1/3. Don't think it affects the substance of anything else in your argument, though, but the math bothered me since intuitively the expected value of originals after any number of steps should be 1, which should be equal to the number of copies in existence times the number of copies. If I'm missing something I'd welcome an explanation of what.