Git add, commit, push from bash script

Question

I'm trying to write a simple cron job that will keep my org-mode files in sync with a GitHub repo by committing every now and then. For some reason, it always manages to commit the files, but even though I see output for the git push when running the script from the command line, like

Enumerating objects: 5, done.
Counting objects: 100% (5/5), done.
Delta compression using up to 12 threads
Compressing objects: 100% (3/3), done.
Writing objects: 100% (3/3), 329 bytes | 329.00 KiB/s, done.
Total 3 (delta 2), reused 0 (delta 0)
remote: Resolving deltas: 100% (2/2), completed with 2 local objects.
To github.com:my/repo.git
   df98efb..0d4943b  master -> master

git status still tells me I need to push:

On branch master
Your branch is ahead of 'origin/master' by 1 commit.
  (use "git push" to publish your local commits)

nothing to commit, working tree clean

Here's my script:

#!/bin/sh

message="auto-commit from $USER@$(hostname -s) on $(date)"
GIT=`which git`
REPO_DIR=~/org
cd ${REPO_DIR}
${GIT} add --all .
${GIT} commit -m "$message"
${GIT} push git@github.com:my/repo.git master

And here's what it looks like in my crontab:

* * * * * echo ["$(date)"] "$(echo "$(~/cron/scripts/org-commit.sh | tail -1)" | xargs)" >> ~/cron/log/org-commit.log 2>&1

In ~/cron/log/org-commit.log, I see this:

[Wed Jun 12 16:44:00 CDT 2019] nothing to commit, working tree clean
[Wed Jun 12 16:45:00 CDT 2019] 1 file changed, 1 insertion(+)

Where the first line was when it run without changes, and the second there was one change. It seems as if git push is never even running, as that log line is from git commit...

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How can I find what's causing this?