I grew mushrooms from spore prints for a while; it is a devilish business when you're working from scratch! Your kit has already done the hard work of inoculating the growth medium with the mycelium; the white fungusy bit you're calling the roots. The mushrooms proper, as already pointed out, are just the fruiting body of the fungal mass.
You should harvest your mushrooms by cutting at the base, disturbing the mycelium as little as possible. Many species will generate multiple 'flushes' of mushrooms before going dormant. Most require a cold snap of some sort to do that, thus the instructions to cool your kit down and prepare for another harvest. Secondary and, if you're lucky, Tertiary flushes will be dramatically lower in yield.
There is no point in leaving any mushrooms to distribute spores and attempt to start another cycle; your substrate is depleted and won't be a viable base for new growth. To top it off, growing fungi is a tricky business. They grow super-easy, but the conditions that are great for your yummy variety (warm, humid, nutritious) are perfect for nearly every other mould and fungi spore floating around, and there are a lot of those in your air.