In answer to your main question: The weeds came from seeds that were in the soil and no affected by the weed-killer applied to above-ground weeds.
An organic gardener's technique for ridding soil of unsprouted seeds and under-soil propagation-capable plant segments (UPPSs): There is a method called solarization that kills occulted seeds and UPPSs and does so without requiring the use of products with "kill" in their names.
In broad strokes, the method uses the heat of the sun, assisted by water and heavy plastic sheeting, to bring about the de-commissioning of the weed seeds and UPPSs that lie in wait no matter how many above-ground weed plants you eliminate.
The following is from Dyck Arboretum of the Plains in Heston, Kansas.
Steps to Solarization
It is best to use this method during the longest, hottest days of summer. The goal is to get soil temperatures under the plastic above 140 degrees. It is easier to reach these temperatures in June through August.
This is a process that will last for a couple months. Plan ahead in your planting schedule so solarization has enough time to work. Some of the more aggressive weeds will not be eliminated in just a few weeks.
- It is best to remove existing growth and lightly till the entire area.
- Remove stalks and debris that will puncture the plastic.
- Rake the area smooth. It is critical that the area is completely flat so plastic lays right on the soil with no air pockets.
- Irrigate the entire area so it conducts heat better. The soil should be moist to 12 inches deep, but not muddy. This is a real trick in clay soils. This is a critical step in the process, because it is not recommended to re-irrigate after the solarization process has started.
- Dig a 8-12 inch trench around the solarization area.
- Lay one entire piece of plastic over the area and tuck the edges into the trench you just dug.
- Cover the edges of plastic in the trench with soil, pulling plastic tight as you move across the whole area. This makes a good seal around the entire site.
Link to source: https://dyckarboretum.org/solarization-for-weed-control/
In your case you might try it on a small area of your lawn this summer. If it works, you could go the whole hog later in the summer or next year.
Optional account of my own experience with solarization:
When solarization is suggested as a means of weed control, the claim that it will be 100% effective is never made. The Dyke article says it worked in some places and not in others. I had a kind of good luck with a rough approximation of the technique. It's optional reading because my installation was idiosyncratic and might not generalize liberally. Still, it should be helpful to some. It's along the lines of "what not to do" in gardens situated like mine.
in 2014, about a year after I'd moved back to a home with a very weedy front yard, I decided to dominate the weeds once and for all. I decided I would grow trees and nothing else. I started by laying down a product called weed cloth with plans to cover it with some kind of bark or gravel and be done with weeding forever. In addition to the cloth and grossly over-priced soil "staples I bought a few sacks of ground-up tires to use instead of bark (decomposes and harbors weeds) or gravel (expensive and painful to bare feet). The tire chunks gave me the creeps because the roads that tires roll over for years are covered with all ids of oil and gasoline-related stuff and in contact with asphalt much of the time. The manufacturer said it was "safe" so I proceeded as if it were safe.
The installation did keep weeds out of my life for a few years. It began to look ratty, though, as the cloth decayed around the edges and the ratio of crypto-dog waste to tire chunk climbed and with entropy being the dogged foe of order that it is. With some courageous helpers, some large trash bags, and the promise of beer, later, I de-installed the weed barrier.
What we found beneath it: It had done more than just suppress weeds. (This is relevant: my neighborhood has clay soil. I forget to mention it at the outset.) Years of mainly dry but sometimes rainy weather minus most of the sun's light plus a moderately weighted fabric covering had produced something more challenging than clay soil: ceramic soil. The result of 5 years in a make-shift whole-yard kiln is pretty close to concrete, actually. If I wanted to dig in it now, I'd start with a hammer drill.
I'd left enough uncovered space around my trees that they were doing fine.
Since the "reveal," I've let the cast-off leaves and sheathes of my 4 or 5 Bambusa oldhammii (the clumping 60-foot true bamboo) carpet the formerly cloth-and-tire covered areas. No one has called code enforcement. I'm calling the bamboo material "mulch." There aren't very many weeds, so maybe this project, in Phase 2, is a success. Or maybe it hasn't rained since 2019.