Switched from Android to iOS today and encountered this issue.
1. This is not Apple's problem.
2. This is not Google's problem.
3. Stop throwing insults when mis/underinformed 🙂
The issue lies within a layer of authentication on the google side. Google account login requires HTTPS/SSL in order to pass information. This step fails silently in many situations on iOS. It was upon manually attempting to add the google account as a generic SMTP account that I found what my issue was, since the application echoes errors instead of running what appears to be a simple script to next/next/finish prompts through the gapps api. I could do an nslookup, ping, all the usual network stuff from my device. I'm getting annoyed with google service features, so downloading the gmail app was not an acceptable answer.
I couldn't do a deepdive due to slight unfamiliarity with how google authentication works, but I did the following:
1. (easy/annoying) Disabled 2-factor authentication. This part is all over this form. This didn't work.
2. (annoying) Brought to factory settings. This didn't work.
3. (real easy) Disabled wifi and went to data. This worked. Why? Because my work was disabling SSL over HTTPS. I confirmed with my helpdesk team that this is the case. Many private sector companies and universities do this. Disconnect from your wifi, or go to a library run by your local government. Or, like, your parents' house or something.
gmail manual settings, in case you want to find out what your error is and diagnose using the internets: https://support.google.com/mail/troubleshooter/1668960?hl=en