I dont know about NZ and Aus, but for you to be a Flight Instructor or getting any flying job in US I'm pretty sure you must have a green card first. You are already in the right option, in today world I think securing a job after flight training is the most important and convinience things to do. Because it will make you easier to do flight training without thinking about getting a job after getting your license. Doesn't matter if its only sitting in jumpseat as long as it pays you for your flight training. And also for SO program in Singapore, they dont state it in their website but its usually SG and MY citizen only who get called for interview.