Honestly, it's not as simple and straightforward as you're making it out to be. Europe is a highly desired location for international teachers and competition is fierce. Without an EU passport you are going to face additionally difficulties.
Credible schools do not want to hire people who have minimal experience. A CELTA is not worth anything in international education. If you want to go to Asia and teach English to locals then China is still willing to hire you if you are white, but this really the lowest on the pole type of work.
There are a number of low caliber schools in China and the Middle East who cannot fill roles easily and may be more willing to hire someone with minimal classroom experience. You do take on the risk of working for an organisation that may or may not pay you, provide professional development or a secure working environment.
However, New Zealand trained teachers are well regarded in the international field because they tend to be flexible, creative, have good classroom management skills, strong inquiry teachers and willing to try new things.