Actually, evolution equations are even more than just second order in time : they don't depend naively on first order derivative, that is, on "velocity". This can be easily understood as the fact that there exists no privileged inertial frames. The change (that is, what is absolute) is given by acceleration and not velocity. If it depended naively on some velocity terms, then it would implies that there's a privileged frame.
Let us make some analogy with Newtonian mechanics. If we were living in an Aristotle universe with privileged frame of reference, then $F = mv$. Motion would therefore be absolute and so would be velocity. Because there is no such privileged frame of reference, but a whole class of privileged ones (the inertial ones), $F = ma$. Why couldn't it be that we live in a universe where $F = m \dot a$ ? Simply because of Galilean principles.
If you believe that acceleration and velocities are "cancellable", and that real change is given by the derivative of acceleration, then you would have to believe in a second order Galilean principle of invariance and inertia. Second order principle of invariance would tell you that the laws of physics has to be the same in all inertial frames and all uniformly accelerated frames, otherwise it would mean that there is a way to discriminate them, and thus, that there is no equivalence between being inertial or uniformly accelerated. This, in particular implies that if you're inside one of these frames and you see someone that is uniformly accelerated with respect to your $x$ axis, that is, $x_1(t) = gt^2/2$, and you also see someone accelerated in the opposite direction, that is, $x_2(t) = -gt^2/2$, then from the point of view of $x_2$, the first object will be described by $x_2(t) = g t^2$. This implies that you would be able to see objects with arbitrary high acceleration, and this without the need to consume any "energy".
This is not what we observe in this universe, you don't uniformly accelerate an object "for free". So it looks like nature choosed to be as simple as possible in order to keep a symmetry between all inertial frames : its second order in time, not third or even worse.
Note that one could say that its Machian, that is, that it is symmetric up to all order in acceleration. This would implies that there is no difference at all between rotation and being inertial. That is to say, that if I look at a guy spinning with a ball in his hands that will eventually let it go, the ball will then make a spiral movement and its angular velocity will keep increasing as far as it goes further from the guy who launched it (indeed, the latter has to see it going into straight line by Galileo principle of inertia). Universe is therefore not Machian either.
Then why does Schrödinger's equation depends on first order in time ? Because it is a modal equation : it needs an observer to makes sense and to make measurement. Hence, there is one Schrödinger equation per observer (the Hamiltonian depends on the observer and the system he is looking at, see the relational interpretations). At least, this is my interpretation of it.