You can pull/dig the Dandelions and Violets out and the Vinca will be fine. You will get more of both coming up from seed but it is manageable.

Those darn wild violets!! My most pesky weed because you have to dig each one out individually and get the entire fibrous root, and even then it probably seeds everywhere as it is ubiquitous in lawns.

I don't think starting over would offer you any real advantage- you'd still have to deal with weeds, and if wild violets are everywhere around you then you will have wild violet weeds.

Some people swear by Preen which would stop the germination of new babies but you'd still have to dig out the existing plants, so a combination approach of the two is your only chance.

If misery loves company please know that I did out piles of these things every day from various locations in my yard. They are my most hated weed.

Violets for me too are really, really a PITA. And digging them out of the lawn is very difficult. All I can attempt is to keep the problem down to a dull roar.