Why can't cell division happen the other way around?
If you are thinking of a process like meiosis but followed by DNA duplication, the problem is that this would create daughter cells that do not have the same genome as the parent cell.

The diploid genome of a sexually reproducing species' cell has different alleles of the same locus. If the cell replicates itself by passing one chromatid to each daughter cell which then duplicate that to make a diploid genome, the resulting cells will not have the same DNA as the original one since each of the daughter cells will have inherited only one of the available alleles. They will then make an exact copy of the DNA they have inherited and become diploid but with a new genome, one that did not exist in their parent. This is not good since the cells of an organism need to have the same genome in order to function in the same way.