maybe it would be beneficial to replace it or strip it down to ensure your customer never has
 to worry about storage performance, for example.

The way the user runs it means that they don't store any metrics locally. The Prometheus agent on the cluster does scraping and then sends the data immediately up to the central SaaS instance (Cortex).

  what if your potential customer runs on a PaaS like Heroku

The range of exporters is very extensive (https://prometheus.io/docs/instrumenting/exporters/). And, as K8s has become so important lots of other people are publishing Prometheus metrics.

  In the end, to offer Prometheus/Cortex as a SaaS and be competitive with existing players
  like Datadog, SignalFX, etc I'd expect you'd have to run an API that your customers can code 
  against to push data to you.

Cortex provides a scalable backend storage for Prometheus metrics, it's designed as one component that someone running at large scale or a SaaS vendors needs. Your comparison isn't apples to apples.

In Weaveworks case we've build integrations into our product that help connect and pull data from other systems. Overall I think Prometheus metrics have a strong future, and there will be lots of SaaS options for sending and consuming the data.