I don’t have a controller icon to enter my Google account

Me and a my friend had this problem (not at the same time). For me, it was just a game glitch, and I reinstalled the game and got the icon back. My friend's phone was quite laggy and didn't have much RAM, and he changed phones and it worked for him. So I guess you could try reinstalling the game, and if that doesn't work, if possible, try using a different device as it could be device specific. 

If I uninstall then all my progress goes away and that's what I'm trying to fix though.

@gamerguy6312 Hasn't your progress already been backed up to a Google account though?

I did what you said to do but instead of putting my age as 12 I put it as 26. I uninstalled but now I have my account registered thanks for the suggestion though.

My office WiFi blocks gaming sites. When I am connected via my office WiFi, I don't get the controller icon.

Once I switch to my home WiFi, everything works properly.