Would you use a load balancer or an application gateway to route traffic based on location? i.e. global store that needs to be accessed from Australia and London and the company wants to ensure a low latency experience.
Or would this better be handled through a CDN?
CDN to start with. That will give you the simple and most effective improvement.
Traffic Manager uses DNS to find the closed connection: https://docs.microsoft.com/en-us/azure/traffic-manager/traffic-manager-overview