In my experience, plants don't like drastic, traumatic changes. Many can survive just fine, but some won't. The safest way to proceed is to prune slowly, giving the trees a chance to adapt over time to new conditions, and form a new shape. Don't lop off everything that carries a leaf and force the tree through more stress than necessary - if something else happens to come along at the same time that you were not anticipating, like a terrible cold snap, an infestation of some kind of pest, a flood, whatevyer... the tree will die because it was already stressed out to the max.

I don't know how to explain exactly how I would prune them, but I have a decent feeling for how much a tree of a certain size needs proportionally to do well. It's easier to see in person. Small mangoes are a little tricky since they hold all their leaves in a concentrated location at the end of a branch, but a safe way to do this would be to cut just one or two branches at a time, maybe a foot back just before a node so it can sprout new leaves fairly quickly. Leave multiple branches in good shape with healthy leaves so the tree is not much affected. Wait until the new cuts have healed and brought forth healthy full-size leaves, and then cut the others. If the tree still looks leggy, you can cut back the same branch further, once it has shown it can heal from the length you already took off. No need to do it all at once for such a small tree - a well-rooted tree can handle more severe cuts, but that thing is tiny still and more vulnerable. It might take you 6 months to a year, but you'll get there, and you won't harm your tree in the process. It's sort of a bonsai situation. You have to be patient. You may find, if the trees produce fruit this year, that this method might also result in larger fruit on the branches that you left intact. That is often the case with selective pruning, because the tree will focus its energy differently. If you decide to keep any of the fruit, you may get a few large mangoes, and then when the season is over and the cut branches have healed, you cut the remaining branches and the ones that were cut first may produce bigger fruit the next season. Then when the tree is finally all evened out, it should have more branches in a more concentrated area so it will be aesthetically more pleasing, and with so many branches functioning at full capacity simultaneously, your fruit should go back to normal. But if you cut out many of the new mangoes in the next crop, the ones remaining will be bigger.

This strategy has worked well for me on maybe 80-100 different trees, and eventually they all get to be a pleasing shape. The health of your tree should always be top priority to its appearance. That way you don't waste time and money growing trees that you then kill all for the sake of tree vanity.   The harder you are on your trees, and the more trauma you subject them to, the less likely they are to survive. Severe pruning - especially leaving no leaves at all - is really taking a major chance that is unnecessary. And if one does not have patience, one should get out of the fruit growing business, right? Sooooo much patience needed for these plants. Plan in terms of years, not months, wherever possible. Prune gradually, and the plants will thank you.