I have the same "problem" with Mouse Eared Chickweed. It's getting to spread everywhere in the lawn. For the first couple of years, I dug it out. But that only slowed it down.
Also, I am getting old enough to where the thought of getting down and grubbing out some random plant in my random garden just because I have some random standard seems foolish. I take Stone's perspective: live with it.
The thing about this forum is, people come in with these "problems" and everyone jumps to help them "solve" it, with some sort of technical advice.
Gardening is a state of mind. There's no one right way to do it. Standards are set by our society and we feel we have to measure up - come hell or high water.
I hope this forum will, along with all the technical advice thrown around, give some thought to why do we even do what we do, and why do we get so upset when things go "wrong" - when gardening could be a joy, with standards we make up as we go along.
So yeah, I did not address the question per se. But I did address the problem.