I have an ASUS ZenPad 3S 10 (Z500M, model P027) and I can't add any shortcut icon from the app drawer to the home screen by pressing and dragging.
I know I used to have this option, but now something has changed and prevented it.
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The issue might be that the launcher layout is locked. Try to remove or replace an icon already present in launcher and if you can't move the icon, then most likely the layout is locked.
To turn off the option, the ways are different in different launchers, try going to launcher settings and find option to lock the desktop layout or something similar and turn it off.
Thanks to everyone, I did not know that it is an app that launches Android out of the root; now I do.
After researching several launcher apps, I got enough understanding to look for the launcher for my tablet. I found it under default apps and it is actually part of the onboard software apps, i.e. resident and native to the hardware, that can not be uninstalled but it has two levels/layers and one of those levels put the app's icon on the bottom drawer so that it can be accessed as I was looking for.