Apps cannot be directly transferred between devices, but must instead be downloaded from App Store to a new or replacement device.
When using QuickStart, the transfer process creates a record of the Apps that are installed on the donor iPad - and it is this record of installed Apps that is transferred to the target (new) device. For Apps that are still available for download from the App Store, the corresponding Apps are automatically downloaded and installed from the App Store. Similarly, if restoring an iCloud or iTunes backup, the backup contains a record of installed Apps.
Not all Apps will, necessarily, continue to be available. Apps may have been removed or retired from the App Store by the respective App Developer. Some Apps may have been abandoned - and may not be compatible with the new device. Where the App is not longer available, you’ll not be able to transfer/install the App.