Uninstall Avast Internet Security completely and reboot to test as disabling it may not be enough to test if its the problem.
Also when using Avast you normally have to “disable it’s built-in incoming mail scanning” otherwise it will interfere with eM Client. See link below.
“eM Client 8 Avast work-around”
https://forum.emclient.com/t/em-client-8-avast-work-around/64297
If disabling Avast incoming mail scanning or uninstalling Avast to test makes no difference to the Gmail connection issue, then try disabling all non OS background tasks on the boot (to test), as their could be some other startup program interfering with eM Client.