Why are large amounts of DNA required for analysis?
Firstly, A DNA sample obtained from a cell or whatever is never going to contain just the DNA sequence you want but will instead contain the entire genomic DNA, plasmids, RNA sequences and whatnot. By using PCR you can selectively multiply the amount of desired DNA sequence while the undesired sequences do not get copied. Keep the PCR going long enough and the amount of undesirable DNA versus the desired DNA sequence is going to become negligible. At that point you can do whatever experiment you want with your DNA sample and be reasonably sure that whatever result you get is caused entirely by your DNA sequence of interest.

Secondly, even if you could isolate a single DNA molecule what would you do with it? You couldn't determine its length by gel electrophoresis as a single DNA molecule wouldn't make a visible band and building it into a plasmid will most likely fail as you only get 1 chance. You probably wouldn't even be able to tell you had a DNA molecule as machines in your everyday lab aren't sensitive enough to detect a single DNA molecule. 

Thirdly, having a large quantity of DNA allows you to perform different experiments from the same sample. Most DNA analysis methods I know of are destructive in that the DNA used cannot be reused for another experiment, so a large enough quantity of DNA to perform all planned experiments without having to obtain a new sample (and introduce all kinds of potential problems) is useful. 

So, in summary, PCR allows us to filter the desired DNA sequence from a sample, perform meaningful analysis on this sequence and lets us use one sample for multiple experiments. There might be even more advantages to PCR that I have missed but these the ones that sprung to mind.