Depending on how you present it you can potentially get people to keep these kinds of writings even if they don't believe it will extend their lives in any meaningful way,

Writing isn't feasible, but lifelogging might be. (see gwern's thread). The government could hand out wearable cameras that double as driving licenses, credit cards, etc. If anyone objects all they have to do rip out the right wires.

In fact it's not even obvious to me that a stack of worlds would "converge": it could hit an attractor with period N where N>1, or do something even more funky.

I'm convinced it would never converge, and even if it did I would expect it to converge on something more interesting and elegant, like a cellular automata. I have no idea what a binary tree system would do unless none of the worlds break the symmetry between A and B. In that case it would behave like a stack, and the story assumes stacks can converge.

I'm really confused now. Also I haven't read Permutation City...

Just because one deterministic world will always end up simulating another does not mean there is only one possible world that would end up simulating that world.

Level 558 runs the simulation and makes a cube in Level 559. Meanwhile, Level 557 makes the same cube in 558. Level 558 runs Level 559 to it's conclusion. Level 557 will seem frozen in relation to 558 because they are busy running 558 to it's conclusion. Level 557 will stay frozen until 558 dies.

558 makes a fresh simulation of 559. 559 creates 560 and makes a cube. But 558 is not at the same point in time as 559, so 558 won't mirror the new 559's actions. For example, they might be too lazy to make another cube. New 559 diverges from old 559. Old 559 ran 560 to it's conclusion, just like 558 ran them to their conclusion, but new 559 might decide to do something different to new 560. 560 also diverges.. Keep in mind that every level can see and control every lower level, not just the next one. Also, 557 and everything above is still frozen.

So that's why restarting the simulation shouldn't work.

But what if two groups had built such computers independently? The story is making less and less sense to me.

Then instead of a stack, you have a binary tree.

Your level runs two simulations, A and B. A-World contains its own copies of A and B, as does B-world. You create a cube in A-World and a cube appears in you world. Now you know you are an A-world. You can use similar techniques to discover that you are an A-World inside a B-World inside another B-World.... The worlds start to diverge as soon as they build up their identities. Unless you can convince all of them to stop differentiating themselves and cooperate, everybody will probably end up killing each other.

You can avoid this by always doing the same thing to A and B. Then everything behaves like an ordinary stack.

Until they turned it on, they thought it was the only layer.

That doesn't say anything about the top layer.

Then it would be someone else's reality, not theirs. They can't be inside two simulations at once.

Then they miss their chance to control reality. They could make a shield out of black cubes.

Why do you think deterministic worlds can only spawn simulations of themselves?

Of course. It's fiction.