common: Rework package-publish-safety-catches.sh to use the CI checks (#2050)

**Summary**

Run the CI checks in the safety catches instead of using a separate
implementation.

Depends on #2038. Resolves #1496.

**Test Plan**

Run `go-task run-safety-catches` on a branch that is ahead of main (with
these changes checked out):


![afbeelding](6e12562c-a1e0-4163-93b8-c2486faf11e9)

**Checklist**

- [x] ~~Package was built and tested against unstable~~

.github/workflows/script_lint.yml

@@ -43,7 +43,6 @@ jobs:
           check_together: 'yes'
           ignore_paths: >-
             common/Legacy/**
-            common/Scripts/package-publish-safety-catches.sh
             common/Scripts/helpers.zsh
             common/Scripts/sync_licenses.sh
             common/Scripts/new-package.sh