The man pages for bash are fairly comprehensive and I'd definitely recommend referencing them while writing scripts: https://linux.die.net/man/1/bash
Why is that? Why don’t they build both from a single source?
This is the theory. The practice is that GNU mandates Texinfo for its projects, and because documentation tends to be the weak spot for all open source projects, manpages end up the most neglected as a result, which can be quite annoying. Especially since pretty much nobody else uses Texinfo - man pages are good enough for most console apps, and those that need more detailed documentation use Docbook, Markdown etc.
Typically (unless this changed at some point) the info pages are the canonical reference for GNU projects and the man pages are to accommodate Unix hackers.[1]
[1] https://www.gnu.org/prep/standards/standards.html#Man-Pages