updater: preserve log files in `updater.log` after an install

Installers such as Chrome and MSI installers can create log files with
the `.log` extension. These log files are useful to examine if an
install fails, so this CL makes changes to preserve the log files within
`updater.log`.

Fixed: 438291028
Change-Id: I5c0d02dfecd6c2c731e70803d204c35ee882cf4b
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6843928
Reviewed-by: Joshua Pawlicki <waffles@chromium.org>
Commit-Queue: S Ganesh <ganesh@chromium.org>
Auto-Submit: S Ganesh <ganesh@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1501635}