Enable playwright tests in deployment workflow

[myieye]

Description

Playwright tests seem to be pretty stable now, so we can reenable them in the deployment workflow/job.

I removed the browser caching steps with the thought that deployment doesn't run often enough for them to add as much value as they do complexity.

Type of Change

• New feature (reenabling workflow step)

Checklist

• I have performed a self-review of my own code
• I have reviewed the title/description of this PR which will be used as the squashed PR commit message
• I have commented my code, particularly in hard-to-understand areas
• I have added tests that prove my fix is effective or that my feature works

How to test

Ran the workflow here and the playwright steps are green: https://github.com/sillsdev/web-languageforge/actions/runs/3322471978

[github-actions]

Unit Test Results

368 tests   368 ✔️  8s ⏱️
  38 suites      0 💤
    1 files        0 ❌

Results for commit 804bc33.

♻️ This comment has been updated with latest results.

[myieye]

Uncovered a flaky test: https://github.com/sillsdev/web-languageforge/actions/runs/3322410002/jobs/5491390263
So, I'll fix that up before merging this.

[longrunningprocess]

Uncovered a flaky test: https://github.com/sillsdev/web-languageforge/actions/runs/3322410002/jobs/5491390263 So, I'll fix that up before merging this.

I just noted this on the other PR, thanks Tim!

[megahirt]

Looking good! I'm very happy to see that we are at this point again with E2E tests

[longrunningprocess]

Testing

GTG on develop run ✅