All the tutorials I've found on the Internet demonstrate how to deploy a web site/app in an nginx container and set up an Ingress to proxy related requests to that container.
That seems kind of redundant to me. Why not to place the content on a volume, mount it to ingress-nginx-controller and serve static assets from there. It seems to be better from performance perspective: no additional proxy and no additional container (with nginx) that consumes compute resources.
An Ingress resource is an abstract concept that was designed to route HTTP/S requests to kubernetes services. It defines a limited subset of that functionality to remain generic and be implemented by many types of ingress controller.
ingress-nginx is only one implementation of an ingress controller, that happens to also be good at serving static content. HAProxy, Traefik or the AWS lb are at the other end of the scale.
An nginx based ingress controller could add custom annotations to support something custom like what you propose, but I imagine you would have a hard time arguing that into the project. Internally, they already add a second nginx instance for the default backend.
If one proxy hop is a critical differentiator in performance then you would probably get more benefit hosting the static content on an edge CDN/device. Another alternative is to not rely on an Ingress and publish the service directly to the outside world.
You need a container to run the application. Ingress service is just used to access that application outside the Kubernetes cluster. You cannot just keep the content in a volume and mount it to the ingress controller. One Ingress controller is used to route requests for multiple applications.
