You can add garbage collection because the Forth program is a language implementation. It probably won't look exactly like ANSI Forth in the end, but that's (theoretically though potentially not in practice) ok.
> It's directly analogous to static type-checking in a 'normal' programming-language, except we use the stack for accepting parameters and for returning values.
I'm familiar with type systems for stack languages, but the issue is that this only really could apply to a standardized Forth language. Forth is only incidentally concatenative and stack-based --- this just happens to have both a simple implementation and has good compositional properties. There is nothing stopping you from introducing new programming models in a Forth, and the article gives a few examples. You can add arbitrary bytecode interpreters to a Forth, for example, and easily make it interoperable with the threaded interpreter in the base Forth. Nothing stopping you from adding words so your Forth feels like a register machine, either. It's because of this that any sort of type system seems doomed (other than one that describes state transitions of the CPU...).
> Perhaps I seem dismissive ...
I sort of don't see the point of saying "it's not really extensible if I can't extend it in all possible ways." In any case, Forth and Lisp are, for trivial reasons, languages that let you embed arbitrary other languages within them (it's about as exciting as how Turing completeness seems to be easy to meet -- which is to say, not very). Worst case, the cost is implementing a whole compiler or interpreter for said language, but, still, it's possible. Common Lisp gives you many hooks to change low level behaviors, so there is usually a better way than this worst case.
Something like borrow checking, though, is a pervasive new feature. The issue is that everything needs to know about how ownership is transferred. It's no different from, even in C, changing some basic struct the whole application uses and then having to update everything to account for it. You could add borrow checking to Common Lisp, but it would have to be in demarcated areas in which borrow checking is being done, and, like Rust, you'd have to figure out an 'unsafe' to be able to use all the Lisp (respectively, C and C++) that's already out there.
> formal methods
Being able to bolt on embedded formal methods to a programming language is active research. I think it's unreasonable to expect this of any language right now, other than ones specifically designed for it :-) (Speaking as a Lean user.)
Forth doesn't really seem to be the kind of thing where practitioners would care about formal methods... It's very defensible to say, then, that Forth is wrong because of this (we depend quite a lot on software being correct!). But, I don't see anything about Forth that prevents you from defining words that check formal specifications -- and I don't mean this in the trivial adding-a-formally-checked-language-into-Forth way.
> extensibility
Anyway, I don't think it's good or bad to be extensible. It's just a property, and Forth and Common Lisp happen to be examples of languages that are much more extensible than usual. There are certainly engineering challenges either way when it comes to extensibility. And, with an extensible language, while you might be able to make bespoke solutions, you now have a bespoke (hopefully small) language to maintain, too. Language design ability and programming ability don't necessarily go hand-in-hand, either...