Productivity Analytics: Replace HAML/JS filtered search with Vue component

On the Productivity Analytics page we currently utilize the filtered search bar which is based on a HAML template and JS. This was introduced in https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/14793

We should replace this with the Vue component which will be implemented in https://gitlab.com/gitlab-org/gitlab-ce/issues/57396 and move all filters to the ProductivityAnalyticsApp.