Add artifacts uploading back to handbook-build-and-review-deploy

Why is this change being made?

I think this build failure happened because of changes to artifact uploading I made.

This run of the same branch passed and didn't have the changes from 70490. I did a quick look through most of the successful builds that happened after my artifact change, and none of them had run the check-handbook-edit-links job, since it only gets triggered by changes to specific files.

I added artifact uploading back to handbook-build-and-review-deploy to see if that resolves the failure. With the changes from 71093 still included, the job was successful.

I rebased my branch so it now only includes the artifact uploading change. @cwoolley-gitlab - you said the original job failure might be flaky variables, but I re-added it to the merge train and it failed again. I'm still new to some of the merge train stuff here. Did I misunderstand what you said or how to check it?

Author Checklist

• Provided a concise title for the MR
• Added a description to this MR explaining the reasons for the proposed change, per say-why-not-just-what
• Assign this change to the correct DRI
  • If the DRI for the page/s being updated isn’t immediately clear, then assign it to one of the people listed in the "Maintained by" section in on the page being edited.
  • If your manager does not have merge rights, please ask someone to merge it AFTER it has been approved by your manager in #mr-buddies.
  • If the changes relate to any part of the project other than updates to content and/or data files please make sure to ping @gl-static-site-editor in a comment for a review and merge. For example changes to .gitlab-ci.yml, JavaScript/CSS/Ruby code or the layout files. (this requirement has been removed pending identification of a new DRI for the handbook)