You should not need to give terminal permissions to modify other applications. 

• What is the result of your script?
• Can you share the script you are using to update the application?

My script fails when deploying it automatically on reoccurring check-in, but it works fine through the Self-Service app or if I enable Terminal to modify or delete applications. The script also works if no previous version of Egnyte is installed, but when trying to update the app to a newer version using the same script, it fails. Here is my script,

#!/bin/bash

#Kill running app
pkill Egnyte

# Download the Egnyte DMG file
curl -o ~/Downloads/Egnyte_0.5.0_2028.dmg https://egnyte-cdn.egnyte.com/desktopapp/mac/en-us/0.5.0/Egnyte_0.5.0_2028.dmg?_ga=2.162962236.103462584.1680192558-1929920014.1680192558

# Mount the Egnyte DMG file
hdiutil attach ~/Downloads/Egnyte_0.5.0_2028.dmg

# Copy the Egnyte app to the Applications folder
cp -R "/Volumes/Install Egnyte/Egnyte.app" /Applications/

# Unmount the Egnyte DMG file
hdiutil detach "/Volumes/Install Egnyte"

# Delete .dmg file
rm ~/Downloads/Egnyte_0.5.0_2028.dmg

# Launch Egnyte.app
open /Applications/Egnyte.app

I dont think JAMF uses Terminal to run scripts. JAMF dumps scripts in /Library/Application Support/JAMF/tmp and JAMF invokes the interpreter directly to run the script. 

Using ~/ in JAMF scripts usually does not work very well as JAMF run scripts as Root. If you want it to go in to the users download folder I would suggest using a variable for that. Granted this would fail if no one is logged in, so coming up with your own temp directory or using macOS's temp directory /tmp (or echo $TMPDIR if you are feeling brave) would be best.