Your chicks will likely destroy the grass
@MysteryChicken . Not that its not worth trying, but don't get your hopes up. Chickens love to scratch, and very little (certainly nothing I've found yet) can survive the damage they can do in a confined space, such as a static run. I'm going to try mint next spring, planted OUTSIDE the run in a protected area, and allowed to spread inwards.
Even so, grass only provides a spongy buffer against the mud, it doesn't address the real problem - water in the run. To fix that, you need to (literally) step back,
@Eva2020. Take a look at the area around your run, with particular attention to the topography.
If your property slopes in a way that directs water towards the run, that's a problem. If you have hard surfaces (roof, patio, etc) nearby that don't allow water to rapidly pass into the soil, that's a problem - because it effectively increases the amount of water falling on the run.
Water will find its own level. You should be considering some combination of berms and/or ditches to divert water away from the run, and to accelerate drainage of what does fall on the run. Some quick work with a shovel, followed by some annual rye (cheap, sprouts fast, will help control erosion till your yard grasses have a chance to grow in next year) are likely all you need to address minor to moderate run off issues.
As to muddy brahma feathers, in warm weather, its not an issue unless they stay wet for extended periods. Once the mud dries, it will flake off - they are (or at least mine are) fastidious birds, and will work at cleaning those feathers. Your concern **IS**, however, quite reasonable because in very cold weather, that wet mud can freeze in their foot feathers, and cause big issues. You want to address that much sooner, rather than later.