I have app service with several scaled out-rules predefined, when load balancer routing my request I want to be able to configure health/readiness probes for my application (not the server itself), is it achievable? If not, what workarounds can be applied?
AFAIK, the Azure app service is a managed service by the Azure platform. It is an HTTP-based service for hosting web applications, REST APIs, and mobile back ends without having to explicitly provision or manage infrastructure. So, I don't think you can have more control of load balancing power for app service.
However, if you directly host your application on the underlying infrastructure like VM or VMSS behind of a Azure load balancer, you can directly set the health probe for your backend instance via probe a port on a backend endpoint that reflects the health of that instance and the application service you are providing. You could get more details from design guidance.
Additionally, if you sick on app service, you still can have multiple choices for load-balancing services. You need to add the Layer 7 load balancers that only accept HTTP(S) traffic like Azure front door or Azure application gateway in front of your app service.
For more references, you could read the overview of load-balancing options in Azure.