Syncing Garmin Fenix 7 with Zwift Cycling

Broadcast Heart Rate from within the settings for the Indoor Biking activity

You shouldn't need to dual record. If your Garmin and Zwift accounts are linked, an activity performed wholly on Zwift will end up back on the watch after synching.

On what device do you run Zwift? What heart rate monitoring device do you use?

Personally, I use a chest strap that broadcast over Bluetooth to my phone running Zwift and to my watch over Ant+. My bike trainer does the same. I've decoupled Zwift from Garmin Connect but left it coupled to Strava. I prefer to have the raw trainer data (power and speed) in Connect than the simulated speed that Zwift does.

I connect trainer and HRM to Zwift on a computer via an ANT+ dongle. I can use any HRM that I have available which include Garmin HRM Pro+, Polar H10, Polar Verity Sense or Marq Gen2 or Fenix 7 internal WHR. No activity recorded on a Garmin device. 

That's is what I logically assumed would happen. But alas, it did not. It posted to Strava. But not Garmin Connect.

This might be the issue, too. I was syncing Zwift to my macbook pro and not an app on my phone.

My heart rate monitor is the one built into my Garmin Fenix 7 watch. 

So what is feeding GC your workout information if it's not Zwift? 

The issue is, I use GC to track my training and caloric expenditure, which feeds into other syncs such as cronometer for calorie tracking purposes. If my workout on Zwift is not syncing to GC, as does all my other training (lifting, hiking, etc), I'm flying dark when it comes to indoor cycling. 

Zwift syncs to the watch and hence to GC

You need to connect your Garmin account to your Zwift account, Then it will do.