Does the type of internet connection (e.g., NAT type) impact the likelihood of encountering the 'Edge of Fate' error, and how can I improve it?

Yes, your NAT type can affect your connection. Ideally, you want an Open (Type 1) or Moderate (Type 2) NAT type. A Strict (Type 3) NAT type can cause connection problems. You can improve your NAT type by enabling UPnP on your router, port forwarding (requires router access and knowledge of Destiny 2's required ports), or placing your PS4 in the DMZ (generally not recommended for security reasons).