Update SciMLTesting QA docs checks

[ChrisRackauckas-Claude]

Ignore this draft PR until reviewed by @ChrisRackauckas.

Summary

• Update root and QA SciMLTesting compat to 2.1.
• Use SciMLTesting's shared public API docs QA path; rendered docs checks are enabled where package docs exist.
• Add or render missing public API docstrings/docs entries where needed and remove stale repo-local public-docs QA where present.

Local Validation

• PASS: timeout 3600 ~/.juliaup/bin/julia +1.10 TOML parse over root/test/docs TOML files.
• PASS: no absolute QA [sources] paths and no test/Project.toml in this repo.
• PASS: timeout 3600 ~/.juliaup/bin/julia +1.12 -m Runic --check <changed .jl files>.
• PASS: GROUP=QA timeout 3600 ~/.juliaup/bin/julia +1.12 --project=. -e "using Pkg; Pkg.test()".
• PASS: timeout 3600 ~/.juliaup/bin/julia +1.12 --project=. -e "using Pkg; Pkg.test()".