[MAINTENANCE] Add public-api check back to CI

[cdkini]

Undoing #9587

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• PR title is prefixed with one of: [BUGFIX], [FEATURE], [DOCS], [MAINTENANCE], [CONTRIB]
• Code is linted - run invoke lint (uses ruff format + ruff check)
• Appropriate tests and docs have been updated

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 80.09%. Comparing base (0ce4ca7) to head (f6f4f86).
Report is 2 commits behind head on develop.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #10449      +/-   ##
===========================================
- Coverage    80.09%   80.09%   -0.01%     
===========================================
  Files          461      461              
  Lines        39995    39995              
===========================================
- Hits         32035    32033       -2     
- Misses        7960     7962       +2     

Flag	Coverage Δ
3.10	67.72% <ø> (ø)
3.10 athena or clickhouse or openpyxl or pyarrow or project or sqlite or aws_creds	?
3.10 aws_deps	?
3.10 big	?
3.10 databricks	?
3.10 filesystem	?
3.10 mssql	?
3.10 mysql	?
3.10 postgresql	?
3.10 spark	?
3.10 spark_connect	?
3.10 trino	?
3.11	67.72% <ø> (-0.01%)	⬇️
3.11 athena or clickhouse or openpyxl or pyarrow or project or sqlite or aws_creds	?
3.11 aws_deps	?
3.11 big	?
3.11 filesystem	?
3.11 mssql	?
3.11 mysql	?
3.11 postgresql	?
3.11 spark	?
3.11 spark_connect	?
3.11 trino	?
3.12	66.35% <ø> (ø)
3.12 aws_deps	45.94% <ø> (ø)
3.12 big	54.57% <ø> (ø)
3.12 filesystem	61.53% <ø> (ø)
3.12 mssql	50.04% <ø> (ø)
3.12 mysql	50.10% <ø> (ø)
3.12 postgresql	54.40% <ø> (ø)
3.12 spark	57.89% <ø> (ø)
3.12 spark_connect	46.24% <ø> (ø)
3.12 trino	52.50% <ø> (ø)
3.9	67.75% <ø> (ø)
3.9 athena or clickhouse or openpyxl or pyarrow or project or sqlite or aws_creds	55.21% <ø> (ø)
3.9 aws_deps	45.96% <ø> (ø)
3.9 big	54.59% <ø> (ø)
3.9 databricks	47.71% <ø> (ø)
3.9 filesystem	61.54% <ø> (+<0.01%)	⬆️
3.9 mssql	50.03% <ø> (ø)
3.9 mysql	50.09% <ø> (ø)
3.9 postgresql	54.39% <ø> (ø)
3.9 snowflake	48.56% <ø> (ø)
3.9 spark	57.85% <ø> (ø)
3.9 spark_connect	46.25% <ø> (ø)
3.9 trino	52.49% <ø> (ø)
cloud	0.00% <ø> (ø)
docs-basic	52.56% <ø> (ø)
docs-creds-needed	52.79% <ø> (ø)
docs-spark	52.20% <ø> (ø)

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[tyler-hoffman]

Kinda stressed about all the new stuff. Some of it seems like it just needs to be added to the @public_api, and some seems like a mistake. But we need this check running again, so I think we should merge ASAP and cleanup after.