Geez! 4.5 inch zoysia. Whoever mows that thing should be slapped...and then of course, FIRED!
That lawn doesn't need compost. It needs to be checked for grubs and treated if necessary. But before that, it needs to be SCALPED and bagged. It should never reach more than 2.5 inches and that would be by the end of the mowing season.
There's no telling what kind of critters, both insect and fungal have taken up residence in that jungle...
"The Poor Fish" circa 1930's: The Poor Fish wouldn't have been caught if he'd known enough to keep his fool mouth shut.