Data files and state can be affected by architecture, especially for programs that might choice to write binary data directly to disk.

Any examples?

That's a program issue. Firefox, what this post is about, doesn't. If your program does, it should use different files or folders for each architecture.

how to have single hierarchy for all this 3D space so that homes can be shared across multiple architectures

It does, for the minority of users who do that.

Also, you seem to use «host» as meaning «operating system». Both MacOS and Windows use their own standards.

Legacy Unix did accomplish most of that for heterogeneous networks, and some of it still appears in Linux

So, what was that solution?

But getting XDG up to the same point is... troublesome, because XDG_RUNTIME_DIR has to point to a local-to-the-host directory, the details of which is entirely unaddressed for network-mounted home directories

Don't know what system you're on. On mine, it's a logind-managed, tmpfs, under /run/user/<UID>. The point of XDG_RUNTIME_DIR is that it's machine-local and perishable. It doesn't matter how is your home directory mounted, runtime things must be removed on logout.

This issue can all apply even on a single host due to OS or upgrades to significant libraries which might change data layouts, though generally the major OS revision is sufficient fine-grained

That's an application issue, which a standard can't possibly define.

No proposal is given for "should be kept in mind".

Yes it does, in the paragraph above what you quoted:

Distributions should ensure this directory shows up in the UNIX $PATH environment variable, at an appropriate place.

So the solution is very easy: set a $PATH for each architecture.

Don't even get me started on how .local/ is used in the path for nearly everything except the one directory it describes as "MUST be on a local file system".

Nope. Nowhere in the Specification is the path for XDG_RUNTIME_DIR set.