Indeed, the "API for Supporting Assistive Technologies" section on that page is outdated. Java Access Bridge is no more and the a11y UNO API has been unpublished a while ago, i.e. it's for internal-use only.
From what I can tell, LO just uses/supports the platform-native a11y APIs by now. At least on Windows and Linux. I I didn't have much to do with macOS a11y code yet, but after a quick research, NSAccessibility seems to be the framework there, and at a quick glance into the code LO uses that.
I'm not good at marketing, an attempt to just update technical information in the existing text would be: replace everything from the start of that section ("Assistive Technology (AT) tools such as ...") up to (and including) the bullet list of what AT tools support Java Accessibility API with something like this:
Assistive Technology (AT) tools such as screen readers and magnifiers, as well as other tools, connect to LibreOffice using the native accessibility APIs of the corresponding platform (Microsoft Active Accessibility API (MSAA)/IAccessible2 on Windows, AT-SPI2 on GNU/Linux, NSAccessibility on macOS).
LibreOffice UI elements and documents expose all the information about themselves and their contents via those accessibility APIs.
An enormous effort has been made to have UI elements and document contents fully support the accessibility interfaces; for example: providing text with attributes and character bounds information, and sending all the necessary notifications and providing names and descriptions of controls.
I'd also remove the "via the message loop" from the second-last paragraph.