There must be a few available solutions, perhaps a breed of grass that is more resistant to trampling?
I also think it would be sensible to protect the edges of the grass (next to paths) as this is where erosion is most likely to occur. A 10cm high board along the edges of the path would adequately protect the immediate edge. Where people step over the edge protection, the erosion will be less significant as the grass can regrow from both sides.
I also wonder how effective it is to put down a rubber mesh where erosion is likely to occur. This allows the grass to grow through the holes, but prevents the damage from footfalls.