On a more serious note, I'm in the same boat with the OP as I had my roads reworked this past summer and I'm going to drill oats this spring when I plant summer plots and will probably plant rye grain this fall. Whether it will work or not I don't know but the seed is fairly cheap and I will already traveling the same roads so dropping the drill into the dirt isn't going to cost that much more.
I think The LLC is on the right track with the broomsedge, the problem is you are looking to stabilize a road that sees light traffic. You are going to have to stay off that road especially when it is wet until the grass is well established. Gravel would be your best bet, but obviously it can be expensive if you are talking a good distance.
I had good luck doing what you are looking to do on a 3/4 mile stretch of road in piedmont red clay by hydroseeding with browntop millet. The fiber mulch in the hydroseeding slurry held enough moisture and nutrients long enough for the millet to establish and once it was established I broadcast a light dose of 19-19-19 a couple times before summer thunderstorms. I just left it standing over the winter and by the end of the second summer I had a reasonable stand of native grasses that volunteered along with about 50% millet. By the end of the third summer there was no millet to speak of and a enough grass to stabilize the roadway for occasional traffic in dry weather. I would only mow it once a year, in the early spring just as the grass starts to green back up.