Use a natural mulch or sand or pea gravel (anything natural and non- treated) and save yourself the grief of worrying about plants there. Most "foot traffic" plants can with stand a light step here and there, not the constant pounding and scraping and falling down of kids playing. IMO that's just asking for ugly unhappy plants.

Is there a low growing groundcover that will withstand foot traffic and not need to be mowed that would be reasonable for under a swingset?

No. Try sand or wood chips.

The rubber playground mulch is ideal for this purpose. In my area we also sell a wood product called play chips that are a barkless, sliverless wood product.

Turf grasses - lawns - hold up to foot traffic better than just about any other type of groundcover and even these won't survive under a swingset.