There is overlap with other questions linked in the comments. But, perhaps the focus of this question is different enough to merit a separate answer. There are at least two distinct but equivalent formalisms of QFT, the canonical approach and the path integral approach. Although, they are equivalent mathematically and in their experimental predictions, they do provide very different ways of thinking about QFT phenomena. The one most suited for your question is the path integral approach.
In the path integral approach, to describe an experiment we start with the field in one configuration and then we work out the amplitude for the field to evolve to another definite configuration that represents a possible measurement in the experiment. So in the two slit case we can start with a plane wave in front of the two slits representing the experiment starting with an electron of a particular momentum. Then our final configuration will be a delta function at the screen representing the electron measured at that point at some later specified time. We can work out the probability for this to occur by evaluating the amplitude for the field to evolve between the initial and final configuration in all possible ways. We then sum these amplitudes and take the norm in the usual QM way.
So in this approach there are no particles, just excitations in the field.
It is important to understand how the eigenfunctions are defined. In particular, here we talk about the eigenfunctions of the hamiltionian, i. e. functions that satisfy
$$\hat{H} \psi(x) = E \psi(x) , $$
where $\psi(x)$ is an eigenfunction (function of space only) and $E$ is its corresponding eigenvalue. In case of the hamiltonian the eigenvalue is also the energy of its associated quantum state.
Now, the total eigenfunction is a function of space AND time. The form of these full eigenfunctions is:
$$\psi(x, t) = \psi(x) \exp{(-iEt)}, $$
and the probability density is $P(x, t) =\psi(x, t) \psi^* (x, t) $.
Now you can see that if one takes a single eigenfunction, then the time-dependence (the exponential part) cancels with its complex conjugate in the probability density.
If one takes a linear combination, e.g. $\psi_1 + \psi_2$, then the probability density is:
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t) } \right) \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t)} \right)^*. $$
Taking the complex conjugate one obtains
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t)} + \psi_2(x)\exp{(-iE_2 t)} \right) \left( \psi_1(x)\exp{(iE_1 t)} + \psi_2(x)\exp{(iE_2 t)} \right). $$
Now there is no reason why the above should be time independent. The exponentials with $E_1$ do not cancel exponentials with $E_2$, therefore one expects mixing terms such as $\exp{i(E_1-E_2)t}$ in the final probability density, and thus it will no longer be time independent.
Best Answer
Any equation claiming to describe fundamental process has to have time. We know that in nature system changes with time. They are not static. For example, absorption of a photon by a hydrogen atom is a dynamic process and we need to have a theory which can predict this dynamical process in time. Since Schrodinger equation is supposed to be an equation describing nature, time needs to get involved somewhere.
The question then is why did Schrodinger choose this particular equation? There is no rigorous argument to that. If you read Schrodinger's original paper, you will see that Schrodinger used Hamilton Jacobi Equation, $H(q,\frac{\partial S}{\partial q})-\frac{\partial S}{\partial t}=0$. He replaced action $S$ by, $$\psi=e^{KS},$$ where $K$ is some constant with the dimension of action (seems familiar?). He then postulates that integral of the left-hand side of above equation should be extremized. This leads to Schrodinger Equation. The factor of time derivative in Schrodinger equation comes from the term $\frac{\partial S}{\partial t}$ in Hamilton Jacobi Equation.
Above procedure may seem, and is, ad-hoc. But he was able to calculate energy spectrum of Hydrogen atom successfully with it. He also wrote a paper in which he sort of gives motivation for this procedure. He used the geometrical formulation of Hamilton Jacobi Equation and argued that quantum physics differ from classical in the same way wave optics differ from geometric optics.
In short, we require wave function to have time dependence if it has to describe nature. The way this dependence was introduced by Schrodinger is little bit shaky but not entirely bogus. I would suggest you to read his original papers if you want to understand his motivation.