Kubernetes Ingress Nginx can do server side includes by allowing Server snippets.
While, traefik doesn't appear to support server side includes or edge side includes.
Breaking up monolithic applications and building systems from microservices requires a solid strategy to integrate output from disparate systems into a coherent experience for the end-user. Integrating at the presentation layer using Edge Side Includes (ESI) for page composition is a practical and elegant solution.
Kubernetes Ingress is an API object that manages external access to the services in a cluster. In simple words, it is an entry point into a cluster, and it routes requests to Kubernetes Services. An Ingress Controller object is required for that, and the most popular are Nginx Ingress Controller and Traefik Ingress Controller.
Nginx Ingress Controller has such options because it is based on Nginx, and it is a web server. In its turn, Traefik was developed to be an HTTP reverse proxy and a load balancer and has no such options.
It is important to mention that both, Nginx Ingress Controller and Traefik Ingress Controller, are used for routing traffic but not for hosting web servers. SSI and ESI are options for a web server, and it is possible to use them in a front-end endpoint in Kubernetes on which the Ingress Controller routes traffic.