This is a complicated question with many answers, most of which will be based on opinions. :)
There are a number of companies that do build products on App Engine, although not many of them make much noise about it. Snapchat and Kahn Academy have historically written about their experiences. I work for Bluecore, and our product is based largely on App Engine.
My very personal opinion about why App Engine is not more popular:
* App Engine Standard (the original product) is just slightly too "weird". There are enough third-party libraries that don't work correctly due to the way the sandbox is implemented that it makes things inconvenient. There was originally no SQL database, so you had to use the Datastore, which is a good product but is also quite "weird."
* It was too early. At the time it was launched, a high level platform that auto-scales your code was unusual and scary. Running stuff in VMs in the cloud was still new. Today the cloud is more familiar, so now some people are willing to try again, calling it "serverless" this time. However, it is too late for App Engine: people have already tried it and decided they don't like it, so they won't try it again.
In short: I have a love/hate relationship with the platform. It is technically very impressive, but it is too bad that it isn't more widely adopted.
I think this article summarizing one developer's experience is a good and balanced description of the pros and cons:
I wrote my own brief comparison of "serverless" versus "platform as a service" that also has some opinions: