Hi FeliksM thanks for the feedback. Principally App Service is a platform for hosting HTTP applications hence why we have such checks in place but ACI does not. Your use case is great feedback for us though and is something we will look into in more detail. Our HTTP Endpoint health check is always running and we report if an endpoint is unreachable. We make theses events visible to help with troubleshooting.
From your latest comment it would appear you were still seeing your container restart, could you either create a support ticket or send me your site name via private message and we can take a look further at our logs please?
Also for reference the Full Windows Server images are also supported, they are much larger however and take much longer to start up as we do not pre-cache those images locally so there are extra layers for us to retrieve. I validated this by building an IIS based image on top of mcr.microsoft.com/windows:2004 this afternoon.