Document exported API coverage

[ChrisRackauckas-Claude]

This draft PR should be ignored until reviewed by @ChrisRackauckas.

Summary

• Add docstrings for exported/public API names that were missing source documentation.
• Add missing exported API entries to the rendered manual pages.
• Add QA coverage that checks every exported name has a docstring and appears in a docs @docs block.

Validation

• Runic.main(... --check ...) on touched source/test files passed.
• include("test/QA/public_api_docs.jl") passed: exported API has docstrings 1/1, exported API is rendered in docs 1/1.
• Direct Aqua QA run passed earlier: Quality Assurance | 17 | 17.
• Full docs build was attempted with timeout 3600 env JULIA_NUM_THREADS=1 julia +release --project=docs -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate(); include("docs/make.jl")'. It logged unrelated executable-example failures in docs/src/examples/neural_ode/simplechains.md and docs/src/examples/ode/second_order_neural.md before reaching the one-hour timeout.