This all depends on what your goal is. If you want a park like transformation, you're doing pretty well on your own, although I can't see how burning is good for anything. My recommendation here is to have multiple kinds/varieties of grasses, not just K31, so you not so vulnerable to climate change, pests or diseases. I would suggest adding some red fescue, Pennsylvania sedge or at least a different variety of tall fescue.
Alternatively, some of us prefer a diversity of life that would be native to your area. This would be a different kind of ecological system that supports many more kinds of life above and below the ground.
My Dad did essentially the same as you, back in the mid 1950s. He cleared the understory of native oak woods and replaced it with sod that he cut from a natural meadow cow pasture 10 miles away. Obviously, there must have been many kinds of plants, including different grasses in it. I was born in 1959 and as I became plant savvy in my teens (early 1970s), I would find many wild flowers native to that understory (not the meadow) growing in what was then almost exclusively Pennsylvania sedge. That old cow pasture was still there (full sun) and when I roamed it I would find little bits of the sedge, but mostly other grasses. My point is that with different kinds of plant materials available, the ones that are best suited will grow best, and be the most pest free, disease free, and carefree for you.