Google Chrome icon disappears from app tray after restarting?

Try opening the Google Play Store app, and search for Chrome. Which buttons show up on the app page? Do you see "Open" and "Uninstall"? Is there an "Enable" button? If you see "Enable," then it's been disabled somehow. I know you mentioned looking to see if it was disabled, but just to be sure--go to Settings>Apps>All, and tell us where you see Chrome. Is it where it should be in alphabetical order? Or is it at the very bottom of the list?

If it isn't disabled, then you could try the following, but be aware that it will reset your homescreens and make them blank, such that you'll have to set them up again. Go to Settings>Apps>All, select "Launcher" (or possibly "Homescreen"), and Clear Data. I think the App Drawer is part of the Launcher, so perhaps by clearing data, it will force the Launcher to repopulate the App Drawer, and maybe Chrome will then show up.

You could try clearing the Google Now Launcher and Chrome app caches in Settings › Applications. Since an app's configuration sometimes is also stored in it's cache, clearing those caches should fingers crossed, force the apps to rewrite your config to their caches and allow the icon to survive through reboots.

That happens on Stock "Launcher" sometimes, and I don't think that there's anything you can do until Google, or the manufacturer releases an update. Until then you can use a different launcher if it really bothers you. Personally, I use Next Launcher 3D and nothing else. Izzy's response seems most feasible to me.