Trying to understand the necessity of orbital approximation or the independent electron model
Setting the stage

Once the Born-Oppenheimer approximation is introduced, we have (first) to solve the so-called electronic Schrödinger equation that could be written as follows,
\begin{equation}
    \newcommand{\el}{_{\mathrm{e}}}
    \newcommand{\elel}{_{\mathrm{ee}}}
    \newcommand{\elnuc}{_{\mathrm{en}}}
    \newcommand{\nucnuc}{_{\mathrm{nn}}}
    \newcommand{\op}[1]{\hat{#1}}
    \op{H}\el \psi\el(\vec{r}\el) = E\el \psi\el(\vec{r}\el) \, . \tag{1}
\end{equation}
Here $\op{H}\el = \op{T}\el + \op{V}\elel + \op{V}\elnuc$ is the electronic Hamiltonian, $E\el = U - V\nucnuc$ is the electronic energy, and $\psi\el(\vec{r}\el)$ is the electronic wave function. The electronic Schrödinger equation, which is a partial differential equation, is fairly complicated. One of the most frequently used methods of simplification of such equations is technique known as the separation of variables.

Separation of variables technique

The separation of variables is aimed to broke a partial differential equation of many variables into a set of simpler partial differential equations of fewer variables. Note though, that the separation of variables is by no means a universal approach: for some equations a solution of the form mentioned above make the separation of variables possible, for others it is of no use.

The most common way of separation of variables for a partial differential equation in a function $f(x_1, x_2, \dotsc, x_n)$ of $n$ variables $x_{i}$ is by writing its solution as a product of $n$ functions $f_{i}(x_{i})$ each of which is a function of one variable only 
\begin{equation}
    f(x_1, x_2, \dotsc, x_n) = f_1(x_1) f_2(x_2) \dotsm f_n(x_n) = \prod_{i=1}^{n} f_i(x_i) \, , \tag{2}
\end{equation}
and then substituting solution of this form into the equation. 
Using algebraic manipulation one could try to break the resulting equation into a set of $n$ independent ordinary differential equations for each function $f_{i}(x_{i})$ and if succeed, one could solve these independent equations and by plugging all $f_{i}(x_{i})$ back into (2) obtain the solution of the starting equation.

Separation of particles coordinates

It is fairly well known that an attempt to separate the coordinates of individual particles in the time-independent Schrödinger equation
\begin{equation}
    \op{H} \psi(\vec{r}) = E \psi(\vec{r}) \, ,
\end{equation}
by making a substitution $\psi(\vec{r}) = \prod\limits_{i=1}^{n} \psi_{i}(\vec{r}_{i})$
will be successful only if the potential energy function can be expressed as the sum of the separate potentials for each individual particle
\begin{equation}
    \op{V}(\vec{r}) = \sum_{i=1}^{n} \op{v}(\vec{r}_{i}) \, ,
\end{equation}
which implies that the particles do not interact with each other. For such a system of non-interacting particles the time-independent Schrödinger equation, partial differential equation of a function of $n$ variables, breaks up into a system of $n$ independent partial differential equations of functions of $1$ variable only,
\begin{equation}
    \op{h}_{i} \psi_{i}(\vec{r}_{i}) = \varepsilon_{i} \psi_{i}(\vec{r}_{i}) \, ,
\end{equation}
where $\op{H} = \sum\limits_{i=1}^{n} \op{h}_{i}$ and $E = \sum\limits_{i=1}^{n} \varepsilon_{i}$. All this is relatively easy to establish, and besides, this derivation is done in many textbooks in great details, so I leave at as an educational exercise to OP.

Separation of electrons coordinates

Electrons, on the contrary, interact with each other, so that a simplification of the electronic Schrödinger equation by means of separation of coordinates of individual electrons won't work. Mathematically, it is the interaction potential $\op{V}\elel$ that will prevent the separation of the electronic coordinates,
\begin{equation}
    \psi\el(\vec{r}\el) \neq \prod\limits_{i=1}^{n} \psi_{i}(\vec{r}_{i}) \, ,
\end{equation}

But what if we can approximate $\op{V}\elel$ with some model potential which will allow the separation of electronic coordinates? Obviously, such an approximate potential should have the form of the sum of one-electron potentials each of which depends only on the coordinates of $i$-th electron as we have already seen. Physically it would mean that electrons do not instantaneously interact with each other, but rather each and every electron interacts with the average, or mean, electric field created by all other electrons. Hence the name of the approximation, the mean field approximation. The corresponding approximate potential can be formally written as follows
\begin{equation}
    \op{V}_{\mathrm{MF}}
    =
    \sum\limits_{i=1}^{n} \op{v}_{\mathrm{MF}}(\vec{r}_{i}) \, .
\end{equation}
The electronic Hamiltonian $\op{H}\el$ is then approximated with the sum of effective one-electron Hamiltonian operators $\op{h}(i)$
\begin{equation}
    \op{H}_{\mathrm{MF}}
    =
    \sum_{i=1}^{n} \op{h}(\vec{r}_{i}) \, ,
\end{equation}
where 
\begin{equation}
    \op{h}(\vec{r}_{i})
    = 
    - \frac{1}{2} \nabla_{i}^2
    - \sum\limits_{α=1}^{ν} \frac{Z_{α}}{ r_{α i} }
    + \op{v}_{\mathrm{MF}}(\vec{r}_{i}) \, .
\end{equation}
And in contrast to eigenfunctions $\psi\el(\vec{r}\el)$ of the true electronic Hamiltonian $\op{H}\el$, eigenfunctions $\psi_{\mathrm{MF}}(\vec{r}\el)$ of the mean-field Hamiltonian $\op{H}_{\mathrm{MF}}$ defined by the corresponding eigenvalue equation
\begin{equation}
    \op{H}_{\mathrm{MF}} \psi_{\mathrm{MF}}(\vec{r}\el)
    =
    E_{\mathrm{MF}} \psi_{\mathrm{MF}}(\vec{r}\el) \, ,
\end{equation}
can be written as products of one-electron wave functions
\begin{equation}
    \psi_{\mathrm{MF}}(\vec{r}\el) = \prod\limits_{i=1}^{n} \psi_{i}(\vec{r}_{i}) \, ,
\end{equation}
where $\psi_{i}(\vec{r}_{i})$ are solutions of the corresponding one-electron Schrödinger equation 
\begin{equation}
    \op{h}_{i} \psi_{i}(\vec{r}_{i}) = \varepsilon_{i} \psi_{i}(\vec{r}_{i}) \, .
\end{equation}

The main problem with the mean field approach described above is that so far we have just assumed that $\op{V}_{\mathrm{MF}}$ exist in one form or another, but we have no idea of how it actually might look like.
There is a well-known workaround, but its a different story.



Note that I ignored spin completely in the discussion above which is, of course, fundamentally wrong, but spin won't make a big difference with respect to the key concept discussed, the mean-field approximation.