Please don't encourage router-makers to add more functionality that risks exposing my network traffic to the internet. I sympathize with the sentiment, but the people that make these devices have proven again and again that they are unable to execute on basic traffic security.

> routers that host arbitrary apps

Servers. You are describing servers.

Edit: upon reflection, rather a lot of people I know don't have routers, either; they use shared internet (e.g. xfinity) or only have mobile plans.