Updates the time tracking specs to use vue-test-utils

What does this MR do?

As a follow up from !44871 (comment 428627317), this MR refactors the time tracking specs:

• Remove css classes that are only used for specs, replacing them with data-testids
• Replace done callbacks with async/await

No changelog added since this is ~"technical debt" with no visual changes

Screenshots

No visual changes

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers
• Informed Infrastructure department of a default or new setting change, if applicable per definition of done

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team