The whole point of that provision in the DMCA was that filing a lawsuit to get a judge to order something to be taken down takes time. If that takes two or three days to happen the damage can already be done by that point if the item in question is allowed to continue being hosted for all of that time.

The problem with the solution (DMCA) is that it also removes the monetary burden of filing a lawsuit (making it very low-risk to just file thousands of these things off per day, never intending to ever followup with a lawsuit) and the sanity filter (the judge).