What are the downsides of having a bigger dead zone?

A bigger dead zone means less of active zone, and the active zone being more distant. More time until the joystick reacts, harder to find the point where it starts reacting (stop -> slow walk), harder to fine-tune the 'active' angle (e.g. normal walk / light jog vs full sprint), and in the end it doesn't fully solve the drift problem as in a worn joystick turn left can be much closer to the 'inert' position than turn right, the joystick stopping at an edge of the dead zone, not in the center.

In the perfect world, there would be no dead zone at all - the joystick would stop exactly in the center every single time, and merest touch would result in weakest non-zero signal. But in the real world, wear, drift, imprecision, even our own shaky hands need to be dealt with, and the dead zone is a crutch, a method to drown all these inaccuracies. The smaller it is the more responsive the joystick, the finer control possible for the player. So it's set as result of in-depth usability studies, a precarious balance, a middle ground between sinking shortcomings of the joystick and providing a good response.

...and if the joystick gets too worn and the dead zone ceases to suffice - oh, this is also a precarious balance, a subject of in-depth business studies. Sales of replacement accessories is a big source of income for console manufacturers, but make the accessory too crappy and you face a class action lawsuit, recall and bad reviews, so the dead zone must be set just so that the drift won't infuriate too many customers too soon.