ci: Try faking a connection in plugin update test

[anomiex]

Proposed changes:

We've had some cases where problematic code didn't trigger during the test because it needs the site to be connected.

PR #35911 suggests a way to fake the connection. We can't really use that directly since we want the code for all plugins, not just Jetpack-the-plugin, but we can copy-paste the relevant bit of code into the test script.

Also, while we're at it, let's run each test with a clean copy of the database.

Other information:

• Have you written new tests for your changes, if applicable?
• Have you checked the E2E test CI results, and verified that your changes do not break them?
• Have you tested your changes on WordPress.com, if applicable (if so, you'll see a generated comment below with a script to run)?

Jetpack product discussion

p1705092068241859-slack-CBG1CP4EN

Does this pull request change what data or activity we track or use?

No

Testing instructions:

• I think you'd have to merge this in a fork, set up the right secrets so the workflow can run, then make a PR that will break upgrades (like what Backup: Add namespace versioning to Helper_Script_Manager and other classes #34739 did maybe) and see if the CI detects it.

[github-actions]

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[zinigor]

I believe we can only really test this when we have plugins that need to be built. Let's try it!