You can do that. Better yet you can use AD to create a scheduled task that runs as System to install and update however frequently you'd like.
powershell.exe -executionpolicy bypass -file "\\Network Share\Saved Script.ps1"
You can create a deployment to run the powershell script from a network location and either run it using your credentials or use AD to create a scheduled task and run it as System.
Once you use the Installer script in my first post to install the apps via winget you just need to schedule a task to run however frequently youd like the PC's to update using the command below which will run the update command as System.
winget upgrade --all --silent --include-unknown
I should also state that there are better ways of accomplishing this and running programs as System is generally reserved for the OS as there is no password or account for System. This should really be done using a Backup Operators account which is what it was designed for. If using System in this way only admins should have write access to these scripts and they should be stored someplace only admins have access to on your network share as this could potentially be modified and used to deploy something malicious.