The 'you change the system by measuring it' is called the observer effect, and it has to do with wave function collapse and not with uncertainty.

Regarding the HUP: you can derive that without referring to measurement apparatuses, by just looking at the momentum and position operators. Let position x and momentum p each have standard deviation dx and dp, respectively, then by the Cauchy-Schwarz inequality

dx dp >= |cov(x,p)|.

The covariance of x and p is

cov(x,p) = E[(x-E[x])(p-E[p])],

where E[A] is the expected value of operator A, and as the commutator of operators x and p is [x,p]=i hbar,

dx dp >= |E[(x-E[x])(p-E[p])]| = |E[xp]-E[px]|/2 = hbar/2,

which is the Heisenberg uncertainty principle.