How to turn dirt track into a softer more enjoyable surface that isn't synthetic

Add cinders or similar aggregate.

Till the inner lane then steamroll it with not too much weight

Is it really a track or is it an oval laid out on a field that has deteriorated to dirt?

In HS 35 years ago, we had a true track that happened to be surfaced with a mixture of stone dust and sand. During a rainstorm, puddles would form but they dried quickly. The trick to keeping a good surface was to scratch up the surface and then roll it every year or two. If you didn't do this weedy grass would grow. The track had a 220 yard straight and both ends that weren't maintained had grass growing on them even though there really was no soil.

We ran races on both cinder and grass/field tracks. Ours was better smoother and flatter, respectively. Of course, it wasn't as fast as early mondo or even plain asphalt that a few schools had.