Wouldn't mowing it stop it from being so visible?
This "herb" shows up once in a while in my garden, never enough for me to notice it and do something about it.
Your best way to get rid of it is manual pulling before it goes to seed. Using chemicals will only take care of the visible plant but won't take care of the seeds that are starting to take root and soon will be maturing.
You’re right, those pics you took there would make me say black medic.
I got the wrong trefoil, should have said lesser hop trefoil, not birdsfoot. They’re very similar looking.
Still learning my weeds. :-D