Yeah...sounds like your grass, and even your weeds, are running out of “food”. You’d probably need to frequently fertilize it throughout the warm season to feed it without the addition of compost. It’s amazing what organic material can do to nutritionally starving plants. My plants that have been growing in composting wood chips for 2 years are growing amazing as the nutrients slowly feed in along with the beneficial fungi. I had a bunch of struggling potted plants that lit up after top dressing with that material and watering some fresh chicken manure into them. My front yard looks a little like yours. The zoysia out there is trying, but the soil out there is about as organic as a twinkie.