Pretty sure you only need to provide health probe request path when your app doesn't respond on the root "/". This can be the case for some dot net app where the routing defaults to "/api", in these cases /healthz can be useful.
I'd suggest you get a known good app working and then figure out the delta between that app and yours. If both don't work then you likely have an infrastructure issue and not a cluster one (eg. an NSG blocking all incoming traffic).
The app i tend to use for this is the Azure Voting App. You can see one of my manifests here: https://github.com/Gordonby/Snippets/blob/master/AKS/Azure-Vote-Labelled-ILB-WarIngress-NetPolicy.yaml
Install nginx again on the cluster with default config a different ingressClassName and see how you fare.
NB: If you're going to use my manifest file, make sure to change line 101 (IngressClassName). Also note that i'm additionally exposing my service via another LoadBalancer on line 90 - this is helpful for debugging an App Problem vs an Ingress problem during development.