E2E: Improve Playwright test patterns in BasePage.ts

[choo121600]

Description

Refactor BasePage.ts to align with Playwright Best Practices.

This PR focuses on improving test patterns only.
It does not change test coverage or behavior.

Checklist

• page.waitForFunction() with DOM queries → locator-based waiting
• page.waitForTimeout() → state-based waiting
• waitForLoadState("networkidle") → wait for specific UI state
• Manual assertions → web-first assertions
• page.evaluate() for DOM manipulation → observe UI state instead
• CSS :has-text() → user-facing locators

Scope

This issue covers improvements in:

• pages/BasePage.ts

Notes

• BasePage.ts is inherited by all page objects, so changes here affect every spec. Please verify that the updated locator still matches the correct element before pr.

Use case/motivation

No response

Related issues

Part of #63036

Are you willing to submit a PR?

• Yes I am willing to submit a PR!

Code of Conduct

• I agree to follow this project's Code of Conduct

[haseebmalik18]

I will work on this.