Kansas City and San Antonio are about 705 miles apart by road. Split the difference and you arrive near Oklahoma City, the city closest to the halfway point between them. That puts roughly 6 hr 15 min of driving on the Kansas City side and 8 hr 50 min on the San Antonio side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Oklahoma City suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Oklahoma City doesn't have what you're after, Dallas and Austin are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Kansas City and San Antonio.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.