If you create a custom image you would need to create a complex health endpoint that is essentially only considered healthy if all the components baked into your image are considered healthy. This gets harder when you are not the author of the sidecar process on which you rely. With a single image it would be easier to run into a situation where the sidecar process (baked into your image) is in an unhealthy state but your container is not restarted because the app itself is not reporting unhealthy status.
Monolith apps can have many dependency checks and it is not really an issue but I get your point. It can become messy. What I have seen gone into sidecars is TLS-termination, caching, authentication, service clients, metrics and logging. Things I would prefer to have in a dedicated proxy layer or in the images.
