[#2285] Update Github Actions for publish-RepoSense preview

[CYX22222003]

Fix #2285

Proposed commit message

Add additional CI checks for publish-RepoSense

publish-RepoSense is our default tool for deploying RepoSense
reports. It would be good to automate the testing of deployments via
this tool before each merge to strengthen our CI/CD processes.

Let's add a script to audit the preview website.

Other information

[CYX22222003]

@sopa301 I decide to move away from docker and build the jekyll environment directlyly on the github actions runner to simulate the environment. Now, it can detect possible empty page due to routing issues.

PR: https://github.com/CYX22222003/RepoSense/actions/runs/13901095753/job/38892784710