Grass should indeed take over if the soil is right for grass. This is my third year embracing the "organic
gardening for the cheap and lazy" article, and the grass is really winning this year. Three years ago, the violet was a close second after Creeping Charlie, with grass a dismal third, just ahead of
dandelions. I tried toxic gick, I tried pulling them all, I tried throwing hundreds of dollars of new grass seed on them. I could have kept doing those things for decades.
1) Get a soil test. I had conifers, raspberries, ferns; everything that would point to acid soil. The soil test came back alkaline. The steps I've taken to reduce pH are really favoring the grass.
2) Know that it will work if you do the right things. This is a years-long process. Trying to get instant results is what gets us into these situations in the first place.
3) Get a soil test. Wait, I said that... anyway, do it.
I would vote against overseeding. If the soil is right for grass, you can watch the existing grass take over on a daily basis. if you have some bare/weedy patches, the grass will just take over next year. Bury the weeds in 4 inches of compost and watch what a happens.