Agree that "tree rings" can be overbearing and unnecessarily formal in an otherwise informal landscape. So I applaud your quest for options. Fine fecues are grass types which will do better in the partial shade that remains under these this tree. Other options are various semi-shade-tolerant groundcover plants, various perennials, etc. Too many to sit here and list, but there are ways to make this work. Finally, it is at least fairly likely that the tree will succumb at some point, so ask yourself the question as to how much $ and effort you want to put into this.

+om

You could give grass a shot, but even shade tolerant grass varieties need near full sun to be full, healthy looking. Growing grass ends up being more of a science than growing conifers. If you head over to the lawn forums and post, they will recommend you get a professional soil test (not big box store DIY test) and see how the Ph, Organic Matter, and micro/macro nutrients look. Your local extension office will do the soil test for about $10-$20 bucks, or you could got to a national lab like Logan labs.

In addition to those variables, the tree is probably consuming the vast majority of the water in that location and the roots may be near the surface, limiting the grasses root zone.

Probably the best solution would be to use some sort of shade tolerant ground cover informally, combined with rocks. I have also seem Moss/Rock configurations under trees that look pretty nice.

lets be very blunt.... if you fail at the most invasive weed in the world.. grass ... i doubt that anything else will work ... under there ...

large trees are super competitors ... especially for water.. under the tree ... its a veritable desert under such .... and though grass may last if it and the tree grow up together.... it a whole other ball of wax ... to start grass after the tree is huge ...

below is a pic of silver maple roots... and if the pungens roots are anything similar.. you should see why it is so hard to deal with roots of a mature tree ..

ken