What’s the design purpose of Gradle doFirst and doLast?

It has to do with extensibility, reuse and avoiding duplication.

For one built in tasks can be extended like:

task CopyAndThen(type: Copy) {
   doFirst {
      println "this is before the actual copy"
   }
   doLast {
      println "this is after the actual copy"
   }
}

The second common scenario that comes to mind is with multi project builds, where you can have a task definition at the top of the project with common behavior:

allprojects {
    task myTask_a {
        doFirst {...}
    }
}

And then the specific projects can extend that.
The task essentially has a list of the Closures that needs to be run and the choice of doFirst or doLast controls to which end of the list the insert goes to.