How do you customize Start Menu pins of an AD user?

I've got a bit of an issue. Windows 11 Pro, and this is an Active Directory user. We use Group Policy, not Intune. Also, right-click is disabled on the Start Menu and Desktop.

I'd assumed that customizing the Start Menu was as easy as creating a .JSON file and importing it via Powershell. That, or applying the .JSON file via Group Policy. It turns out you can't apply a .JSON file using Group Policy.

Currently, the Start Menu pins are default, so they link a bunch of random Windows apps. None of the existing Start Menu pins are of use, so I'd like to remove them all and create my own.

I could not find a working solution to remove the pins, as I tried a Powershell script. I also couldn't add any shortcuts.

I want to avoid the use of third-party software. How can I customize the Start Menu?