That makes sense but what confused me is this statement on the link that was provided above:
To receive SMS, MMS, or push notifications from third-party apps on your Apple Watch Series 3 (GPS + Cellular), your paired iPhone must be powered on and connected to Wi-Fi or cellular, but it doesn't need to be nearby.
Since email is "Push" i can see why the phone has to be on but then the statement on the Apple site referenced third-part apps and I wouldn't consider the native email app to be considered third party.
I also tried the watch today with my home wifi with the phone off and emails didn't come in either. This surprised me. too. I have had the Watch 3 since launch and never noticed this email requirement since I typically keep the phone on when it isn't with me. This week the phone ran out of battery and I was expecting the watch to be able get emails.