Utterly false. Staying alive has nothing to do with React Native or Cordova or any particular framework. It's about implementing the native APIs that allow that behaviour.
I'm the author of a couple of modules that allow your RN app to operate in the background:
Thank you for linking those modules. I'm in the early stages of an app that will benefit from that functionality and admittedly my knowledge is based off of the first google result of a search for "react native background worker/task" (http://stackoverflow.com/questions/35376690/how-can-i-run-ba...). Which of course makes me an expert on the subject in this day and age.