I like clover in lawns.
When I put down weed killer I try to avoid the area of the two lawns I do that have clover but up North the spurge can be such a problem, and I simply no longer spend hours pulling it by hand, that sometimes I have to hit that area which is hard on the clover, but I have never totally eradicated it .
It seems to spread slowly to not at all though.
Down South after I do a major weed killing, I paid the price for not treating for weeds last year, I am going to try seeding some again next year at the same time I replace some sod I put in that was weed infested and part of my current problems.
Sod, you get what you pay for , OFTEN, be picky .
As an aside: I though moss needed acidic soil. Down South we have acidic soil and until they put in the osmosis system , acidic water, so to see it there was not a surprise, though it was uncommon.
Up North we have , water loaded with calcium and non-acidic soil but I have removed far more moss than I ever expected.