Refactor(issues analytics): move filter into vue

What does this MR do?

This MR refactors the legacy filtered search to be initialized inside the issues_analytics.vue component as preparation to migrate it to use the gitlab/ui filtered search.

Pulling it in here will make it easier to decide whether to use the legacy version over the modern version with a feature flag, as it is done in Analytics > Code review.

Screenshots

There is no visual change

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

There is very low risk with this (small) change. It "just" moves some functionality to happen inside a Vue component, rather than before it is loaded.

• 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

Related to #220344 (closed)