The problems of randomness, whether as a philosophical construct or an aesthetic tool, become clear most especially when you're trying to make it. In computer graphics, one finds rather quickly that simulating randomness in the world (or a world) isn't a matter of rolling dice; dice-roll randomness looks, well, artificial. Instead, we might use something like Perlin noise that, while involving a lot more thought and planning, nonetheless feels more properly random, like in nature. Which isn't that random at all.
We can say that randomness is more of a perspective than a thing itself, a way of describing reality in relation to how much information you have about it. That feels safe enough, but it still doesn't kill randomness as an idea. Is it possible for a thing to have no patterns, no information about how it came to be?
Moreover, while randomness might not be a proper feature of the universe, we've made it one of our techno-reality. Cryptography and computer security, for example, depend on the generation of random numbers. And there are then casinos and lotteries and jury selections and the entirety of statistical theory. Looking at randomness is how scientists are able to look at noisy data and find the patterns within; randomness might be a false idea, but we still experience it even just as a perspective, and so we need it.
Randomness is a deep topic and this is a short post—this American Scientist piece from earlier this year does it some actual justice—but just consider the question, "Is it always possible to find order in a chaotic system?" Ramsey Theory, described in the relatively easy-going, equation-free lecture above (given as part of the Millennium Mathematics Project), offers a reasonable way of determining the minimum amount of structure in a given system.
Basically, if you have some chaotic system, and you start dividing it up, you'll come up with an "interesting" structure. Something will eventually repeat or have some sort of form (order). Curiously, while it guarantees the existence of this form, it offers no way of finding it aside from just brute force (going through and looking at each combination individually until you hit gold). Make of that what you will.