I’m thinking you may have Empire Zoysia, standard turf zoysia for Central Fla.
All this year and since at least last Nov/Dec 2016, I’ve been using a lawn sweeper/manual rake almost weekly for a similar yard w/very large oak tree. Acorns were particularly heavy in Nov and Dec but the extended drought has increased much more leaves on top of lawn. Yet the zoysia does very well – rotors on lawn were nozzled down from 3gpm to 1.5gpm a few years ago and that didn’t make any dry spots and also made some economy on water usage.
Again, mandatory raking of leaves or investing in a lawn sweeper may be something you should consider.
Big trees are very good for shade on a lawn in summer, especially with increasing warming temps in summer. But of course too much shade will hinder zoysia growth which likes lots of sun. In any case, IMHO it is worth keeping one big tree in a strategic area over a lawn, if you are fortunate to have one already, as long as you handle the maintenance and periodic trimming where appropriate. It can be great asset to a healthy lawn.