[ChrisRackauckas-Claude]

Ignore this draft PR until reviewed by @ChrisRackauckas.

Summary

• Update QA to SciMLTesting 2.1 and remove repo-local QA [sources] path entries.
• Use SciMLTesting public API documentation QA instead of repo-local public-doc scanning.
• Add missing public API docstrings and rendered docs entries where this repo has docs.

Validation

• timeout 3600 env GROUP=QA JULIA_NUM_PRECOMPILE_TASKS=1 /home/crackauc/.juliaup/bin/julia +1.10 --project=. -e 'using Pkg; Pkg.test()'
• timeout 3600 /home/crackauc/.juliaup/bin/julia +1.10 --project=. -e 'using Pkg; Pkg.test()'
• timeout 3600 /home/crackauc/.juliaup/bin/julia +1.10 --startup-file=no -e 'using TOML; ... TOML.parsefile(...)'
• timeout 3600 /home/crackauc/.juliaup/bin/julia +1.10 --startup-file=no -e 'using Pkg; Pkg.activate(temp=true); Pkg.add("Runic"); using Runic; Runic.main(["--check", repo])'
• git diff --check