Try putting down some seed mulch or seed mats to help prevent your seed from washing away. Lesco sells a seed starter mulch that has a bit of starter fertilizer and a tackifier in it. The mulch helps to retain some extra moisture and prevent water from washing away the seed. If there is enough water flow, the mulch can also be washed away. In that case, I'd look at one of the straw mat type products. Just stay away from anything with a plastic mesh in it.

Great. Sound like you did not make the common mistakes. The problem out by the street is that the concrete and/or asphalt bathes in the sun all day absorbing light and converting it to heat. It stores the heat during the day and releases the heat all night. With 24/7 heat, the water in the soil evaporates faster there. The grass that was there died from drought. Then one week in the past, you got enough rain on a daily basis to sprout the weed seeds in the bare spot. Some of those weeds likely died for the same reason, but those which did not are likely doing quite well. To fix that problem you need to water more deeply next to the road. By watering deeply the roots will grow deeper where the temperature of the soil is cooler and evaporation is reduced. Measure your sprinkler output with cat food or tuna cans placed around the yard. Time how long it takes to fill all the cans. That is your starting point for watering. If you have some cans that filled right up and others that took significantly longer to fill, then you might need to adjust your sprinkler heads. The worst case is that you might have to replace a couple heads with higher or lower output heads. That's not a big deal, but you need to figure out which ones, if any, need it. The time it takes to water a full inch is your starting point for future watering. Apply the water and watch how long it takes before the grass starts to wilt. This time of year, with mature lawn and "normal" deep roots, it should take at least 2 weeks for the grass blades to fold together or the grass to lay down when walked on. As soon as you see those symptoms anywhere in the lawn, water again right away. If it happens before 2 weeks is up, then water longer next time. When your daytime temps fall into the 70s on a regular basis, you should be watering deeply once every 3 weeks. Below the 70s and you can back off to monthly watering. Then in the spring, bring your watering frequency back up as needed. With temps eventually above 90 degrees F, you will be watering weekly. Why do this? To get deeeeeeep roots. Deep roots get their water from deeper in the soil and can go much longer without surface watering. Also you want to break the life cycle of weeds. Weed seeds need to be kept moist for several days to germinate. If you are watering every day, then you are creating perfect conditions for weed seeds. By watering deeply once and letting the surface dry completely, the weed seeds will never get enough moisture to germinate...that is until Mother Nature intervenes. But at least you don't have to contribute to your own weed problems by watering too often.

I've used a seed mix called "Fleur de Lawn". It's a mix of lawn daisies, and dwarf grasses. In the summer it needs to be mowed maybe once a month. I've let it go all year in some places, without creating a wilderness. If you visit the link I included, click on the "ecology mixes" in the menu on the left, and then "fleur de lawn".

I thought the flowers looked like Tradescantia (Spiderwort,) but didn't recognize the short foliage. After reading again, I see that it was mowed. Don't all of the Commelinas have 2-petaled flowers? Tradescantia is very happy in 1 corner of my yard and I mow around the clumps of it, which get bigger every year. In a couple years, I think it will have taken over that corner and I won't have to mow there anymore. It looks so pretty when it blooms, and the blooming period is very long.