In the time-dependent Schrodinger equation, $ H\Psi = i\hbar\frac{\partial}{\partial t}\Psi,$  the Hamiltonian operator is given by

$$\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2+V.$$

Why can't we consider $\displaystyle i\hbar\frac{\partial}{\partial t}$ as an operator for the Hamiltonian as well? My answer (which I am not sure about) is the following: 

$\displaystyle H\Psi = i\hbar\frac{\partial}{\partial t}\Psi$ is not an equation for defining $H$. This situation is similar to $\displaystyle F=ma$. Newton's second law is not an equation for defining $F$; $F$ must be provided independently. 

Is my reasoning (and the analogy) correct, or is the answer deeper than that?


If one a priori wrongly declares that the Hamiltonian operator $\hat{H}$ is the time derivative $i\hbar \frac{\partial}{\partial t}$, then the Schrödinger equation 
$$\hat{H}\Psi~=~i\hbar \frac{\partial\Psi}{\partial t}\tag{1}$$ 
would become a tautology. Such trivial Schrödinger equation could not be used to determine the future (nor past) time evolution of the wavefunction $\Psi({\bf r},t)$.  
On the contrary, the Hamiltonian operator $\hat{H}$  is typically a function of the operators $\hat{\bf r}$ and $\hat{\bf p}$, and the Schrödinger equation 
$$\hat{H}\Psi~=~i\hbar \frac{\partial\Psi}{\partial t}\tag{2}$$ 
is a non-trivial requirement for the wavefunction $\Psi({\bf r},t)$.   
One may then ask why is it then okay to assign the momentum operator as a gradient
$$\hat{p}_k~=~ \frac{\hbar}{i}\frac{\partial}{\partial r^k}~?\tag{3}$$
(This is known as the Schrödinger representation.) The answer is because of the canonical commutation relations
$$[\hat{r}^j, \hat{p}_k]~=~i\hbar~\delta^j_k~\hat{\bf 1}.\tag{4}$$
On the other hand, the corresponding commutation relation for time $t$ is 
$$[\hat{H}, t]~=~0, \tag{5}$$
because time $t$ is a parameter not an operator in quantum mechanics, see also this & this Phys.SE posts. Note that in contrast 
$$\left[i\hbar  \frac{\partial}{\partial t}, ~t\right]~=~i\hbar,\tag{6}$$
which also shows that one should not identify $\hat{H}$ and $i\hbar \dfrac{\partial}{\partial t}$.
