How long have you had this device?
Factory resetting your device shouldn't cause you to loose any apps and most data is or can be backed up through Google Backup. You will probably have to sign-in to things again but other than possibly some game saves you shouldn't loose much, if anything. Not that I'm suggesting that at this point, just saying.
Do you have a screen protector on this device?
I'm assuming that you've cleaned the screen as well or it's not dirty?
You say touching the home button is the only way to wake it before the screen responds, which sounds like something is running. So like if you had an app open you would have to close the app or go to your home screen to get out of it before you could do the next thing. This brings a few things to mind;
Is it only the home button, or can you get the phone to respond by pressing the back button too? Does this mean that you cannot pull the notification bar down as well?
Have you ever just started at the top of the screen tapping left to right like a typewriter to see if any other potions of the screen respond?
Have you checked the settings for an accidental touches setting?
So does it do this every time you open the phone? If you open the phone touch the home button to open an app, close the phone immediately and open it back up, you still have to touch the home button first or will it remain responsive for a short time, until it has been sleeping for awhile?
Two things I would try after checking for the answer to the last question are clearing the cache partition.
Follow these steps. Then check and see if anything has changed.
If not I would check this behavior in Safe Mode.
Follow these steps. Then because you tested if it happens every time you lock and unlock the phone, you'll know how to recreate the conditions that cause the issue and will be able to check it while in Safe Mode. If it stops happening in Safe Mode there's a good chance that it's being caused by an app. Finding that app may not be easy. You could, while in safe mode open each app individually, then close it, then try to reproduce the issue based on what you learned locking and unlocking the screen. When you start having the issue it would be one of the apps you just closed because in safe mode apps cannot auto start. You have to first open the app.
While trying to work through all of this one other question occurred to me, are you using a launcher besides the default app launcher?