You hit one point on the head: "it’s really just a math problem." But what that means is that it is not a philosophy problem. It was originally posed as a problem in philosophy, but the only thing such considerations do is obfuscate the math problem. Without justification. Just the need to put it in the realm of philosophy to match the original intent.

Consider a variation of the problem, that isn't a variation at all, but I'm sure will get a response as to why it doesn't fit in with a desired philosophical solution. I know this, because it has happened before. Use the same details with four volunteers, but one coin, and different schedules. One volunteer will be left asleep on Tuesday, after Heads, as in the popular[1] version. Each of the other three will be left asleep on a different one of the other three possible combinations of the day (Monday or Tuesday) and the coin result (Heads or Tails).

So on each day in the experiment, three of the volunteers will awakened. Each will be asked, individually, for the probability[2] that this is her only waking[3]. Then, the three will be brought together and asked to discuss the same question. The only taboo topic is the combinations where each would be left asleep.

In the individual interview, each is solving an problem that is identical to the popular version. In the combined one, each knows that she has a 1/3 chance to have the same assigned coin result as the missing volunteer. The only possible issue, is why should the answer change between these interviews.

--

[1] It isn't the original, or even the version posed in the paper that created the popular version.

[2] "Odds" means a ratio, not a probability. Odds of X:Y mean a probability of X/(X+Y). Halfers say the odds are 1:1, or a probability of 1/2. Thirders say the odds are 1:2, or a probability or 1/3.

[3] This is the question as it was in the original version. It is equivalent to "What is the probability that the coin result means you will be wakened only once?"