Perhaps sapodilla.

I think this is fairly prevalent in tropical landscaping, if it's the one I'm thinking of the old leaves turn reddish before falling.

Maybe try tropicals forum elsewhere on this web site.

That's wild looking. It could be a fig/ficus, but it also puts me in the mind of a magnolia. Trouble with either is the habit ~ straight-out limbs spaced so far apart doesn't fit with any cultivars of either fig or magnolia I know of, but there may be one out there that grows that way? Even though you said there were many that way in out of the way places, could it be that there's a "Guerrilla Gardener" pruning them?