It seems that a reasonable usage of --exechook-command / $GITSYNC_EXECHOOK_COMMAND is to run a git-archive command on the local repository after it has been synced from the original source (for example, it's a very convenient way to build a backup).
See the git-archive manpage: the --format= of such an archive is usually some very classical tar, or zip, or tgz.
Thus, it would seem helpful that the git-sync container image provides the tar, gzip and or zip (ideally all of them). Especially since it is hard to add them afterwards. For example, when building a custom image based on git-sync the environnement is already minimal, with no package management tool ready to use, and very few dynamic libraries.
Another option would be to continue having a basic git-sync image, but offer a complementary image with busybox tools on top of it, more tailored to the needs of those who run exec hooks and need a little bit of tooling.