FWIW I agree that the Rust way is nicer, but I can't impose the Rust way on Python. I guess the secret hack is to use PyO3, which pdoc supports quite well. ;)
Let me try to explain why it is that way: First, it's consistent with Python functions. The docstring for functions is below the signature as well. Second, consider a file with only a docstring and then a variable declaration. Here it would be ambiguous if that's the module or the variable docstring. Finally, this behavior is consistent with other tools (and failed standardization efforts) in the space. So yeah - I share your sentiment, but I think it's the most pragmatic approach for pdoc. :)
I'd say neither fork as made great strides since then, but I'm also biased here as the maintainer of pdoc.
There is no pdoc-specific library for link checking as far as I'm aware. It's all plain HTML though, so you can use a more general tool like https://lychee.cli.rs/. :)