Mary, I'm no expert but I do have a nice mango tree that blooms and fruits here in my garden. My thoughts would be it has something to do with the very low humidity in AZ and also maybe being in a pot at 10yr. old is holding the older one back. At age 10, my tree was 15ft. tall and wide. Way too big to be in any pot.
Anything you can do like spraying the leaves with the hose or installing a micro-sprinkler system on a timer to mist the trees to increase the humidity around them might help. Mangoes grow in tropical humid places in nature.
The other thing you're going to encounter, even if they do bloom is lack of pollination. Here the main pollinator for my mango tree is a certain type of fly. If you don't have these flies where you are, it's possible that even if the tree blooms, it won't give you fruit. Don't mean to be discouraging but I'm not at all sure that hand-pollination is possible as the flowers are in large panicles but they are very tiny.