Instruction
In order to use G Suite email with FreeScout you need to create an App Password for the G Suite user.
According to G Suite support App Passwords will work forever - this authentication method will not be deprecated in the future. App Passwords do not belong to "Less Secure Apps" and are not related to them.
1. Allow G Suite users to turn on 2-step verification
As admin go to G Suite "Security » Basic Settings"(https://admin.google.com/example.org/AdminHome?hl=en#SecuritySettings:flyout=basic) and tick the Allow users to turn on 2-step verification checkbox:
2. Turn on 2-Step verification for a G Suite user
Follow this instruction. You can use same phone number for different G Suite users.
3. Create an App Password for a G Suite user
Follow this instruction.
Go to your account's Security, click "2-Step Verification" and there at the bottom you'll see App Passwords section.
Now you can use this App Password as a Password in FreeScout's Sending and Fetching settings.
Troubleshooting
If your FreeScout can't connect via IMAP anymore: imap_open(): Couldn't open stream {imap.gmail.com:993/imap/novalidate-cert/ssl}
Sometimes Google can ban your server's IP address - see this issue. In this case contact Google support.