Why does it have to have been a "ring" of tiny particles? They don't leave impact craters, anyway. Another scenario would be that an asteroid broke into pieces when it came within the Roche limit, but those pieces orbited multiple times before individually impacting Earth. Jupiter did that to a comet not long ago.
The "ice age" may or may not be related. If there was enough ejecta from the impacts, would we have had cooling effects from dust thrown into the atmosphere by the impacts? Could that have been a "tipping point" in Earth's climate balance, rather than a sustained cooling by the dust?
Finding a lot of craters 400,000,000 million years old suggests to me that they must have been substantial impacts to have produced evidence that survived Earth's weathering processes for that length of time. And, those are just the ones that hit the continents - how many hit the oceans?
For that matter, why would it have to be an "asteroid"? Why not a second, smaller moon that was created at the same time as the Moon we all know, love, and want to get back to? Maybe that second moon's orbit slowly lowered while the Moon of today's orbit slowly rose, as is currently happening with the 2 moons of Mars? When the smaller, lower moon finally got within the Roche limit, it broke apart and eventually led to multiple impacts, maybe with a ring.
Just because a theory might fit the sparse data does not make it a sure thing.