Can I answer my own question? I am thinking the logging issue is more so related to Windows 2012 R2 user profile. See the attached Microsoft article for more details and how to work around it. And if you find a hotfix anywhere, let me know.
A team member in the past was getting logs from a location like you pointed out in the first post.
There would be a tabcmd.log in that folder.
But later he created a "service account" to run certain scripts and perform certain tasks.
And started to use task scheduler to run those scripts.
Later when two other coworkers moved on, and I took over Tableau administration.
I ran into the predicament of knowing what scripts were completed or the details of the failures.
Something could "complete" and show as such in Task scheduler but it would not perform as expected and it would produce an error to screen or other but I was not there to see. For example during a backup script early in the AM.
When looking at task scheduler in Windows 2008. We noticed it took arguments.
Or that we could redirect what would be output to the console to a custom log file.
This is starting to be more and more important as tabcmd and tabadmin log files grow and we have more custom scripts for different things.
So we just created a folder for logs and funneled the output for whatever script was being run to an appropriately named log file like the argument below:
we are doing the redirect of the log via the task scheduler command like just like you showed. But it won't show the internals of what tabcmd is doing. For this, you have to go look at the tabcmd.log file.