What you're describing is proxy voting. It works well in certain circumstances: primarily in shareholding, where there are well-understood ways to divvy up authority. But most proxies are given with general power. This makes them comparable but not identical to a representative system.
The problem with proxies for politics is twofold. First, there is the business of carving out subjects. If I hand John the power to vote on transport and Jill the power to vote on buses, who votes on the integrated schedule for subway-bus terminals? What happens if they both vote? How do I maintain anonymity under these conditions without getting double counted, possibly even cancelling myself out through my proxies? This problem doesn't arise in regular proxies because I can see how my proxy voted and anyone can see who was acting as my proxy. That's not compatible with ballot secrecy.
The second problem is the usual problems of direct democracy, chief among which are demagoguery and mobs. Representative democracy is sluggish compared to direct democracy. That is a feature, not a bug. Imagine if twitter mobs could amend legislation.
This is a nice response, thanks. The one miscalculation here is it's clear that representational democracy is also rife with demagoguery. Because the system is 'slower', it gives those in power the ability to cause damage to the system, but keep their office because they support other populist sentiments. Often the issues that are most important are the ones that get abused, while the popular-but-less-important issues are the ones that people vote on for their representative. Without the representative, then these less popular topics can be voted on justly/rightly by people that need these bills (minority issues in particular).
I agree though that the speed of direct democracy can also be problematic in a similar way: too hasty decisions on the popular topics.
The problem with proxies for politics is twofold. First, there is the business of carving out subjects. If I hand John the power to vote on transport and Jill the power to vote on buses, who votes on the integrated schedule for subway-bus terminals? What happens if they both vote? How do I maintain anonymity under these conditions without getting double counted, possibly even cancelling myself out through my proxies? This problem doesn't arise in regular proxies because I can see how my proxy voted and anyone can see who was acting as my proxy. That's not compatible with ballot secrecy.
The second problem is the usual problems of direct democracy, chief among which are demagoguery and mobs. Representative democracy is sluggish compared to direct democracy. That is a feature, not a bug. Imagine if twitter mobs could amend legislation.