migration assistant

Do both your Macs use the same user account name and password? As you mention above, Logic Pro X requires "permission to open application", and this points to the possibility that because the app file was imported from the TM backup, the OS installed on Mac Mini is simply not authorized to run it (i.e. doesn't have the required access permissions). It's relatively easy to correct:

Right/Control-click on "Logic Pro X" file (it's a "package" in fact, so it's similar to a folder), then select "Get Info":

This should open the properties for the app package folder. At the bottom of the properties window, you will see the permissions section which looks like this:

Permissions for "System" and local account have to be "Read & Write", so if permissions in your case are different, you can set them manually. To do that, click on the lock icon at the bottom right corner of the properties window. You'll be asked to enter the name (usually it's already there) / password for your local account:

After unlocking access, select "Read & Write" as permissions for "System" and local user account:

Click on the lock icon once again or close properties window. That's it, you should be able to launch LPX.