Do they? They could probably do more but I'm not sure they think it's all sorted here. Southgate has clearly indicated it hasn't been sorted here and even the FA suit they got out immediately after the game yesterday acknowledged that we have plenty to back at home in this regard, at every level of the game.
I will happily have a pop at the game's authorities in this country, and as I say, we could probably do more. But I don't think they consider we are past all that.
I also don't think that because we still have those issues that we can't call racism out when it occurs. If we had to wait until we'd eradicated racists from the English game before we acted on that of others we'd never do anything. Which gives racists a free pass.