Align build process with upstream

Context - gitlab#441760 (comment 1781149674)

We should align with the upstream build process to have no/minimal deviations.

Note that this is a reversal from the conclusion we reached in this long discussion thread:

gitlab-web-ide#306 (comment 1746320459)


See also this discussion about whether we should use a node image or build node:

gitlab-build-images!780 (comment 1782651315)

Edited by Chad Woolley