"But the real significance of this is that you cannot write a program that can tell you meaningful facts about the behavior of an arbitrary Turing machine. For any program you write to determine the behavior of HTML + CSS, I can write a piece of HTML + CSS that your program will be unable to say anything about."
Sorry, but this is false and for similar reasons is not reasonably considered Turing complete. The encoding presented here produces, for any particular HTML + CSS, a finite number of cells of an automata (and not only that, the number of cells is linear in code size). That means that deciding any property about a given piece of code amounts to search over a finite space.
This is very different from true Turing completeness where a small set of rules might require unbounded space and computation to decide, and in general will not even be decidable. Now, the author of this code says in a parent post that we can't build a real Turing machine. That's true, but doesn't mean that we can't specify a real Turing complete process; it just means that for some executions we have to give up on running them.
In some cases, it makes sense to consider languages that are not actually Turing complete as approximately Turing complete. For example, if you take the C programming language, with a 64 bit memory space, then again there's technically only a finite space of possible program states. But in that case, you can have a 100 line C program that can generate 2^64 possible program states. Turing completeness is a reasonable approximation. If you had to allocate every bit of memory explicitly with a unique line of C code, it would no longer be very reasonable to consider C approximately Turing complete.