When using a Xfinity gateway router does it need to be set in "bridge" mode?

Bridge mode would be if you decided to use better hardware, eg; if you bought an Eero Mesh system, you'd probably want to put in in bridge mode, but if you're using Wi-Fi from the Xfinity gateway directly, you don't want to put it in bridge mode.

That seems to be a weird Xfinity specific issue. You should try sending a test email from another email account to test that your email is working when the situation arises, and changing the setting appropriately if you need to.

https://www.xfinity.com/support/articles/wireless-gateway-enable-disable-bridge-mode

As you can see at the link above, enabling bridge mode probably isn't what you want.