The lawn experts are going to want to know where you live to be able to give you great advice. And I believe the weed may be Purslane.

I live in Painesville, OH. It's on the east side of Cleveland.

That last pic looks like spurge with the red stems.

That's actually a nice job. Did you use Tenacity? The color looks a little pale for elite cultivars. You have plenty of coverage-- i would not overseed if i were you and just let it mature some more with liberal fall fertilzer. Watering is tricky. We've had decent rain. Conditions are currently mild. Root system will be shallow so water accordingly. Keep an eye out for fungus and be prepared to treat immediately. Your job is really to keep it alive until fall where you will really see the lawn take off.

I used Scotts starter fertilizer which has the same active ingredient as Tenacity.

Nice. The lack of weeds is really impressive. Only thing I would do different is keeping up with the milorganite applications until september and then switching to synthetic.

Yes that is unfortunate timing, and yes, that is a pretty good stand of lawn without weeds especially considering the timing. KBG is the kind of grass that spreads to fill in thin spots. Usually there is no need for reseeding or overseeding unless there was some serious insect or disease damage in the summer. KBG is also the kind of plant that sleeps the first year after seeding. They say the first year it sleeps, the second year it creeps, and the third year it leaps.

Watering first season grass can be tricky especially since it went in in the spring. The roots are delicate and easily damaged by the heat. Are you watering longer now? Have you done the tuna can test to see how much water your sprinklers put out? I see you are using an impact sprinkler on a tripod. Those are nice in theory, but you will get much more even coverage with a gear driven oscillator sprinkler. If you go with one of those, then 10 or 20 minutes a day is not nearly long enough to help. You might need to go to an hour a day to get about 1 inch per week. As soon as the daily high temps fall back to the 80s, then start to back off on the frequency and go up on the time. I would try to go to twice a week at about 4 hours (oscillator) each time. Generally the tuna can test is to see how long it takes to fill the cans, but for you I would leave the sprinklers on for an hour and measure how full the cans are. The idea is the same; to determine how long it takes to fill the cans with an inch of water. That will eventually be your watering time every time. The only thing that changes is the frequency which changes with the cooling temperatures. As you back off on the watering frequency you should see the grass becoming darker and darker green, which is why you selected those cultivars of KBG.

Here are a couple updated pics.

About 3 weeks ago I put down another application of Scott's Starter Fertilizer because i had it laying around. I haven't actually sprayed tenacity as of yet. I figured i could get my dosage in that way. I do have weeds and I haven't noticed a ton of POA Triv or Annua but the weeds I have mostly the Pursaline noted above.

I do have a few POA Annua and Triv a little bit. What would be the best route to go? It has been 3 weeks since i put the last Scott's starter fertilizer down. Is it too early to do a dimension application if I still want my grass to spread?

As long as you don't overdo the dimension, your grass will spread just fine. Even when I was using pre-emergent, I was still seeing sporadic poa annua however. The lawn probably won't be super dense until well into fall so you are vulnerable until it is. I have no experience with the starter + mesotrione but it obviously appears to be doing a great job. That said, going forward, I think at some point you will need to use the spray form of mesotrione. I think anyone with an elite KBG lawn needs it to control grassy weeds as an ungoing cultural practice.