You didn't tell us if this is a USB  cable from the printer. You have it right, but at "turn on the printer" you should wait a while. The printer will warm up and initialize, and then it sends a communication to the computer, which, as PnP (plug and play) will detect the device. Then, you wait while the computer tries to understand if it already has what it needs to set up the device (windows has a lot of device drivers in it, now). Usually, when you download anything nowadays, it's just a control panel for the printer, for things such as "out of paper" notifications or access to more advanced functions.

For instance, you PnP the USB. Then, you want to activate the WiFi function, so that is additional steps for the user. Also, there might be multi-function, such as scan-and-email, that you would set up.

Just give the computer enough time to do the hard stuff first.

By the way, (and I might be cursing it), the printer I still use is a Duplex Brother from 2009. It still works with that same set of steps as you describe.