Why do plasmids replicate on their own?
In plasmids with high number of copies, the plasmid needs a certain density to be able to supply its function. If a given cell contains a few dozens of plasmid mollecules, and this is its optimal density, the easiest way to achieve this is to downregulate the replication by its own presence or the presence of one of its components. This also allows a random segregation during cell division, due basically by simple diffusion. It's also easier for the plasmid to achieve its optimal density once it has enter in a new cell by conjugation or transformation.

Plasmids are also subject to selection, so it is reasonable to assume that the more independent and more contagious a plasmid is, the bigger its presence in the population. In fact, it is a reasonable hypothesis to assume that many viruses are parasitic plasmids (many viruses in fact have a similar genomic structure and share many traits with them).

However, in plasmids with low number of copies the replication events can be dependent of the genome replication as you describe (despite the fact that many plasmids of this kind still regulate its own replication by itself). In this case, though, the line between plasmid and chromosome is less clear. In some organisms some big plasmids seems to form part of the genomic organization in the whole strain, and they usually contain important genes. This plasmids behave just like regular chromosomes, and they segregate by the same mechanisms the rest of the genome use. For those plasmids the terms minichromosome and megaplasmid are used, and the choice usually depends of the kind of genes it contains.