Most prefer to graft their mangoes over layering. I would be really interested to see Steve's mango layers. When I've purchased from him in the past, they've all been grafts.
Lack of a tap root shouldn't be an issue. They can be a little less resistant to the wind I suppose? Most of the mango trees you buy from a nursery are going to have a badly malformed taproot anyway -- you'd need to grow them in a tree pot (long and narrow) to allow the taproot to develop.
The main reason mangoes are not layered is because it's far less efficient for mass production. Grafting is quicker and easier. Plus, you can produce hundreds or thousands (using budding) of grafts from a single tree vs a few dozen using layering.
Another common reason nurseries will prefer layers over grafts is when the tree has a tendency to sucker from the rootstock (eg, guava). However, a good nursery will skip the layering and just use tender cuttings (Pine Island, for example), which are faster and more efficient.
That said, I would suspect that layering would be best in the fall (Sept), as that's when mangoes send a lot of energy into root production.