Answer: Don't get rid of those kids yet -- you just need to plant zoysia grass. It is very wear- and shade-tolerant, and much better adapted to your warm climate. Fine fescue is a good choice for overseeding to keep your lawn green through the winter, as long as the kids can play somewhere else, because it doesn't wear well. Fescue fades with the heat as the zoysia takes over. It takes some time for zoysia sprigs to completely fill in an area, but it's worth waiting for. The blades are coarser than fescue, so it's not as kind to bare feet (but that may keep it from being too trampled!). Contact your county extension office for the varieties of zoysia recommended for your area.
Since the kids need a place to play, consider converting the area to lawn in stages. Give the kids a piece of the yard as their own, and spread bark mulch there to protect the soil underneath from excess compaction. When the zoysia comes into its own, let the kids use the turfed area for their play, and convert their play area to lawn. You'll probably have to fluff the soil a bit before planting sprigs there.
You also have a great opportunity to landscape parts of the area with groundcovers such as ferns, mondo grass (Liriope), lilyturf (Ophiopogon), or woodland natives. These don't take traffic, but can add another dimension to what sounds like a wonderful park-like area. Good luck!
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