All lawns, including chemical free, require some TLC. Top photo, lawn is overwhelmed with weeds and bare spots.
This can be result of low nutrition, compacted soil and blowing that leads to low microbe levels. Lawns like this can occur after stopping chemicals after years of chemical treatment because chemicals negatively impact soil microbes important to soil health. Blowing is another problem because important nutrients are blown off of surface. Restoring lawn is really about restoring soil and soil microbes.
You can have a much improved lawn by following steps here: https://www.dirthacker.com/post/organic-lawn-hacks-your-fall-to-do-list-early-fall
Nets out to:
Application of organic fertilizer mid April & Sept. (take care not to over fertilize). Lawns are hungrier than most weeds.
Get a soil test from local extension, see blog link above for details.
Cut at three inches and leave lawn clipping in. Cutting at three inches is very good way to reduce weed competition.
Water judiciously, only when dry, and deeply. Frequent light watering or watering when not needed harmful.
Reseed in fall--mid Sept. is good in this region. Choose a good seed mix that is mostly Tall Fescue because Tall Fescue is most heat & drought tolerant.
Mow leaves in in the fall. Any modern mower will mulch leaves well
A chemical free lawn example follows. Note there is some weed, but some weed is a good thing.