chore: Move public-api-check script to GH action

[shibeshduw]

The check public-api-script is converted to a GH action with inputs. This is then reused in the AI SDK (PR).
Closes https://github.tools.sap/AI/gen-ai-hub-sdk-js-backlog/issues/115

[shibeshduw]

NOTE

Ignore the many junk commits. I couldn't get nektos/act to run actions locally on my machine, so I had to do some trial and error here.

The public api check works as intended both here and in the AI SDK.
The only problem here is with the unit tests for the public api check. This test case passes in one build/tests action but fails in the other two build/tests actions (see the checks above). According to the current logic, the current code is correct and the test should pass in every case. The issue arises with exportAllInBarrel() not identifying the barrelFile that has been mocked in the test.

[deekshas8]

It seems existsSync doesn't work consistently between node 18 and 20/22.

[deekshas8]

LGTM