test: server.getUrl is now a promise

[fredericoo]

WHY are these changes introduced?

Fixes intermittent issues locally when the server URL is requested before it had time to start, and if it didn’t have a hardcoded port it would throw an error

WHAT is this pull request doing?

• server starts in constructor instead of separately (we always created a new instance and initialised it at the same time anyway)
• server.getUrl() now is a promise, so we can await for it to be done in our test statements and then we have the base URL to go to

HOW to test your changes?

• run e2e locally
• check CI in this pr

Checklist

• I've read the Contributing Guidelines
• I've considered possible cross-platform impacts (Mac, Linux, Windows)
• I've added a changeset if this PR contains user-facing or noteworthy changes
• I've added tests to cover my changes
• I've added or updated the documentation

[shopify]

Oxygen deployed a preview of your fb-playwright-server-stability branch. Details:

Storefront	Status	Preview link	Deployment details	Last update (UTC)
Skeleton (skeleton.hydrogen.shop)	✅ Successful (Logs)	Preview deployment	Inspect deployment	March 5, 2026 1:42 PM

Learn more about Hydrogen's GitHub integration.

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