The whole point is that it isn't fragmented. How you learn now is fragmented. In order to defragment you have to fill in the gaps between those bits of knowledge. Without knowing how it fits in or where it comes from you can never truly understand what it is.
Children do not have the self disipline or maturity to recognise or care that their knowledge is fragmented or not....in most cases they will simply not care and by the time it registers that it might adtually be important it will be too late.
There is no way you can state that allowing a child to learn whatever he want will naturally create a structured unified pathway to knowledge......you will have to go a long way to convince me that your average kid will choose to have anything to do with Algebra, Statistics, Geometry, Physics, Chemistry or any number of subjects other than the bare minimum needed to do whatever it is they want to do.
In most cases you will have a generation of illiterate poorly educated 16 year olds wondering what to do now......
Also teaching is not fragmented, curriculums are designed to give a structured pathway to each subject.....learn to walk before you can run basically.
Whether GCSEs are stringent enough or whether the emphasis on exam results is right or wrong, you cannot seriously propose that allowing children to learn what they want, when they want will actually be an improvement.....
Kids will simply not do what you are proposing.....they will just mess about, choose the easiest pathway and learn as little as they can get away with so they can play Crysis and gossip about the neighbours daughters boobs.....
Children simply do not have the self discipline, determination, foresight, maturity and basic skills required for that kind of freedom......that is what university is for....once you have learned how to learn of your own violotion and have the skills, maturity and basic knowledge to do so effectively.
And that is the reality, not your rose tinted interpretation of it.