Consolidate file writing boilerplate into the _render function

This code was repeated several times. Moving it to a common location
simplifies the code.

Bug: 330885075
Change-Id: Ia4365fda63a4d6194c165bc298ec70c10caa1925
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5392101


Reviewed-by: Andres Ricardo Perez <andresrperez@chromium.org>
Commit-Queue: Jean-Philippe Gravel <jpgravel@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1282149}