Update View Multiple Times Within Function

You need to execute porcessData in another thread

// Where you call ProcessData():

        DispatchQueue.global(qos: .userInitiated).async {
                self.ProcessData()
          }

Now, as we call UI functions in processData, they must be called from main thread

    func processData() {
        DispatchQueue.main.async { //  as we call UI function
            self.statusText.stringValue = "Message 1"
        }
        read_data1()
        DispatchQueue.main.async {
            self.statusText.stringValue = "Message 2"
        }
        read_data2()
        DispatchQueue.main.async {
            self.statusText.stringValue = "Message 3"
        }
        read_data3()
        DispatchQueue.main.async {
            self.statusText.stringValue = "Message 4"
        }
        read_data4()
        DispatchQueue.main.async { 
            self.statusText.stringValue = "Done"
        }
    }

Note:

- names of func should start with lowercase

- convention is to use camelCase more than underscore: read_data1 -> readData1

- don't forget to prefix with self in Dispatch blocks

- I assumed the name of IBOutlet is statusText, not statusMessageLabel

     @IBOutlet weak var statusMessageLabel: NSTextField!

You could also Dispatch in the func itself

If so, just call processData() in code directly, without enclosing in a Dispatch.

change func as:

    func ProcessData() {
        DispatchQueue.global(qos: .userInitiated).async {
            DispatchQueue.main.async { //  as we call UI function
                self.statusText.stringValue = "Message 1"
            }
            self.read_data1()
            DispatchQueue.main.async {
                self.statusText.stringValue = "Message 2"
            }
            self.read_data2()
            DispatchQueue.main.async {
                self.statusText.stringValue = "Message 3"
            }
            self.read_data3()
            DispatchQueue.main.async {
                self.statusText.stringValue = "Message 4"
            }
            self.read_data4()
            DispatchQueue.main.async { 
                self.statusText.stringValue = "Done"
            }
        }
    }

I tested, with a dummy read, it works OK.

If it works for you, don't forget to close the thread.

And to answer your first question:

What is the best/accepted way to relay the changing status to a user?

Dialog?

TextField inside the window?

HIG make it clear that you should not abuse of alerts:

Minimize alerts. Alerts disrupt the user experience and should only be used in important situations like confirming purchases and destructive actions (such as deletions), or notifying people about problems. The infrequency of alerts helps ensure that people take them seriously. Ensure that each alert offers critical information and useful choices.

So, in your case, TextField seems most appropriate (as you said you don't want to use progress bar: why ?).

Thank you for the response, Claude. I will be testing your suggestions tonight.

I wouldn't mind a progress bar. I've never used one, so I know I should learn.

I didn't want to use a "spinning thing" because if a user is unsure how long it should take, how long would they wait until they Force Quit?

Even if the updates are fairly quickly, I want the user to see activity.

Excellent! Status and ProgressBar implemented and working.

Thanks Claude!