AA is not an attempt at reparations, it is an attempt at integration. In law school admissions, for instance, minorities are referred to as under-represented. The intent is to give them a leg up on integration, not to make up for some past transgression.
In order to collect actual reparations, one would have to prove an economic loss. The claimant would need to identify individuals that were wrongly enslaved, tally their hours of labor, find the prevailing labor rate, subtract off their cost of living (which considering the subsistence nature of the work, would likely barely exceed the cost), and then apply interest until the present day. Furthermore, the claimant would need to identify the actual owner of the slave and pursue the debt from their family. All of this is irrelevant of course, because it is a basic principle of law that one cannot be charged retroactively for a an action that was made illegal after the fact. Therefore, there is no ability to collect against a party for an action that was not considered a tort at the time of commission. One might say, "well then it is the governments fault for being negligent in providing an adequate framework of laws". This is of course, also irrelevant since our society and legal system rests on the individual. The government, even if it were somehow responsible, cannot compensate a nebulous group of society. Only individuals can collect. In order to do so, they must prove their case.
The only part of the original argument that I have discussed here is that of lost labor. Actions (or inactions) of the government during the civil rights movement are a different issue, since one could argue about damages done by the government negligently not enforcing laws. Still though, each individual, in order to be compensated monetarily, would need to prove an economic loss.