A few years ago I wrote an App-V recipe for SSMS 18.5. For MSIX, the older SSMS versions (up to 19/20) also worked well: unpack the package and fix the permissions of the registry keys in User.dat, as described in the Tech Community article [FIXED] Azure Virtual Desktop (AVD) – Packaging the SQL Server Management Studio (SSMS) to MSIX. In the MSIXForcelets this is Add-MSIXFixSSMS.
Then Microsoft switched SSMS to the Visual Studio Installer. SSMS 21 and 22 are built on the Visual Studio shell – and the old recipe does not work anymore. The capture with the MSIX Packaging Tool works, the package installs, SSMS starts – and then this: