Is teaching English a good career for a non-native speaker?

Yes, teaching English can certainly be an excellent career for a non-native speaker, and you will not necessarily be unable to find rewarding, challenging work as an English teacher just based on your first language. You might have some options that are closed to you, as mentioned before, as many employers in some parts of the world specify native speakers only. This is changing though, and more and more employers specify  level of English i.e. C2, rather than insist on English being the language you were raised speaking.