Remove `update-tag` event and use existing click event, do not access vuejs elements directyl instead use `querySelector`

https://gitlab.com/gitlab-org/gitlab/-/merge_requests/36753#note_383620360
https://gitlab.com/gitlab-org/gitlab/-/merge_requests/36753#note_383620356
https://gitlab.com/gitlab-org/gitlab/-/merge_requests/36753#note_383620351