Because if you agree that the correct way to measure the probability is as the occurrence ratio along the path, the degree of splitting is only significant to the extent that it affects the occurrence ratio, which in this case it doesn't. The coin toss chooses equiprobably which hotel comes next, then it's on to the next coin toss to equiprobably choose which hotel comes next, and so forth. So each path has on average equal numbers of each hotel, going forwards.

When we speak of a subjective probability in a person-multiplying experiment such as this, we (or at least, I) mean "The outcome ratio experienced by a person who was randomly chosen from the resulting population of the experiment, then was used as the seed for an identical experiment, then was randomly chosen from the resulting population, then was used as the seed.... and so forth, ad infinitum".

I'm not confident that we can speak of having probabilities in problems which can't in theory be cast in this form.

In other words, the probability is along a path. When you look at the problem this way, it throws some light on why there are two different arguable values for the probability. If you look back along the path, ("what ratio will our person have experienced") the answer in your experiment is 1000000:1. If you look forward along the path, ("what ratio will our person experience") the answer is 1:1 (in the flaming-tires case there's no path, so there's no probability).

I don't know what I meant either. I remember it making perfect sense at the time, but that was after 35 hours without sleep, so.....

The answer to the second part is no, I would expect a 50:50 chance in that case.
In case you were thinking of this as a counterexample, I also expect a 50:50 chance in all the cases there from B onwards. The claim that the probabilities are unchanged by the coin toss is wrong, since the coin toss changes the number of participants, and we already accepted that the number of participants was a factor in the probability when we assigned the 99% probability in the first place.

You're reading a little more into what I said than was actually there. I was just remarking on the change of dependence between the parts of the problem, without having thought through what the consequences would be.

Now that I have thought it through, I agree with the presumptuous philosopher in this case. However I don't agree with him about the size of the universe. The difference being that in the hotel case we want a subjective probability, whereas in the universe case we want an objective one. Subjectively, there's a very high probability of finding yourself in a big universe/hotel. But subjective probabilities are over subjective universes, and there are very very many subjective large universes for the one objective large universe, so a very high subjective probability of finding yourself in a large universe doesn't imply a large objective probability of being found in one.

The most obvious difference is that the original problem involved the smaller or the larger set of people whereas this one uses the smaller and the larger.

I fail to see why that is the general case.

If you have two people to start with, and one when you've finished, without any further stipulation about which people they are, then you have lost a person somewhere. To come to a different conclusion would require an additional rule, which is why it's the general case.
That additional rule would have to specify that a duplicate doesn't count as a second person. But since that duplicate could subsequently go on to have a separate different life of its own, the grounds for denying it personhood seem quite weak.

For that matter, I fail to see why losing some(many, most) of my atoms and having them be quickly replaced by atoms doing the exact same job should be viewed as me dying at all.

It's not dying in the sense of there no longer being a you, but it is still dying in the sense of there being fewer of you.
To take the example of you being merged with someone, those atoms you lose, together with the ones you don't take from the other person, make enough atoms, doing the right jobs, to make a whole new person. In the symmetrical case, a second "you". That "you" could have gone on to live its own life, but now won't. Hence a "you" has died in the process.

In other words, merge is equivalent to "swap pieces then kill".
The above looks as though it will work just as well with bits, or the physical representation of bits, rather than atoms (for the symmetrical case).

If you mean that a quantitative merge on a digital computer is generally impossible, you may be right. But the example I gave suggests that merging is death in the general case, and is presumably so even for identical merges, which can be done on a computer.