How/Why are linear systems so crucial to computer science?
The premise of the question is a little flawed: there are many who would argue that quadratics are the real "boundary" for tractability and modelling, since least-squares problems are almost as 'easy' as linear problems. There are others who'd argue that convexity (or even submodularity in certain cases) is the boundary for tractability. 

Perhaps what is more relevant is "why do linear systems admit tractable solutions ?" which is not exactly what you asked, but is related. One perspective on this is composability. Since the defining property of a linear system is that $f(x + y) = f(x) + f(y)$, this imparts a kind of "memorylessness" to the system. To build up a solution to a problem I can focus on individual pieces and combine them with no penalty. Indeed, the premise of most algorithms for flow is precisely that. 

This memorylessness imparts efficiency: I can break things into pieces, or work iteratively, and I don't lose by virtue of doing so. I can still make bad decisions (c.f. greedy algorithms) but the act of splitting things up itself doesn't hurt me. 

This is one reason why linearity has such power. There are probably many others.