I am curious what type of data you are trying to pull from the Tableau designing dashboards? Like usage statistics? I might be able to find some better resources with a better understanding of the desired result.
Tableau offers some insight into the Tableau Server data and the workbooks on it: Administrative Views - Tableau
There is also the ability to create custom admin views: Create Custom Administrative Views - Tableau
Additional information on connecting to the PostgreSQL repository: Collect Data with the Tableau Server Repository - Tableau
I did find some other resources where people are capturing error messages with PowerShell: Tableau-Powershell-Scripts/Tableau-Server-Backup at master · glenrobinson72/Tableau-Powershell-Scripts · GitHub
Hope this helps!