In spite of the fence, and the do-over, you'll likely need to repeat the do-over every several years.

Violets fling their seeds to 8 or more feet away from parent plant.
Stuff blows in.
Birds drop stuff.
People and animals trek in stuff.

No way possible to have a pristine lawn.

Selective weed-killers are available to kill all weeds you've mentioned because they are broadleaved plants. Grass is a narrowleaf plant

Or start over, then remain vigilant for the few as they appear.

Agreed. Upon further investigation, I think you'll find that it's not good (rot) for a wood fence to touch the ground. A barrier that extends from below the surface up beyond the bottom of the fence would be easier to maintain and the fence should last longer.

Well I just looked at your pic to see what specific observations could be made and realize it's not a flower bed you're trying to protect. You don't have to subscribe to the "a lawn is one type of grass with nothing else growing in it" idea unless you want to. Good luck either way!

Those seeds from those unwanted plants will be brought in to you yard from places far beyond you next door neighbors and will land and germinate in your yard.
You can spend a lot of money, time, and energy "getting rid" of those unwanted plants, poisoning the world you live in while doing that, or work at growing a good, healthy turf that can aid in preventing these "weeds" from growing.