Add a Playwright e2e test

[msabramo]

Write a Playwright test that tests the app e2e.

Motivation and Context

Ensure things don't get broken

How Has This Been Tested?

abramowi at marcs-mbp-3 in ~/Code/OpenSource/inspector (playwright-test●)
$ npx playwright test client/e2e/transport-type-dropdown.spec.ts

Running 1 test using 1 worker

  ✓  1 … Type Dropdown › should have options for STDIO, SSE, and Streamable HTTP (764ms)

  1 passed (1.5s)

Breaking Changes

No

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

Additional context

[msabramo]

Related PR (which has perhaps fallen into some disrepair): #354

[olaservo]

Thanks for setting this up. My draft PR I'd opened for feedback has been getting pretty moldy. A couple questions about this e2e setup:

• Are you thinking these should just be run locally during development? Originally I assumed that we should have these run as part of CI so we're not relying on that. I think it would also be OK to work that in as separate PRs.
• I'm making a similar assumption with adding Playwright configs - my original PR had some settings for browser, timeouts, etc. which we can add as we need them.
• How do you think we should approach adding tests for different transports? In my original tests I was duplicating a lot of code.

Thanks again for working on this, it should help bridge some gaps with testing basic stuff efficiently.