Storage: Test mode 2 dual writer in playlist tests

[suntala]

What is this feature?

[Add a brief description of what the feature or update does.]

Why do we need this feature?

I missed updating the playlist tests when I added the mode 2 feature flag in #87668. That means the playlist integration tests are currently testing mode 1 behavior. We don't have mode 1 enabled so we should prioritise testing mode 2 behavior. This is a bandaid PR since mode 2 is enabled in dev and ops and the other modes aren't enabled anywhere. We will need to introduce equivalent tests for all the modes though. Will create a subtask/issue to track this.

Who is this feature for?

[Add information on what kind of user the feature is for.]

Which issue(s) does this PR fix?:

Fixes #

Special notes for your reviewer:

Please check that:

• It works as expected from a user's perspective.
• If this is a pre-GA feature, it is behind a feature toggle.
• The docs are updated, and if this is a notable improvement, it's added to our What's New doc.