First, consider the preference relation $\succeq$ defined by
\begin{equation*}
x \succeq y \text{ if } x_1 + \cdots + x_n \geq y_1+\cdots+y_n
\end{equation*}
This preference relation satisfies your assumptions. Indeed, suppose that $x \succ y \succ z$. You can check that $y \sim \lambda x + (1-\lambda) z$ for a unique $\lambda$ defined by
\begin{equation*}
\lambda = \dfrac{y_1+\cdots+y_n-(z_1+\cdots+z_n)}{x_1+\cdots+x_n-(z_1+\cdots+z_n)}
\end{equation*}
And these preferences obviously admit the utility representation $u(x)=x_1+\cdots+x_n$.
Now, consider the lexicographic preference relation $\succeq$ defined (I take $n=2$ for simplicity) by
\begin{equation*}
x \succ y \text{ if } (x_1>y_1) \text{ or } (x_1=y_1,x_2>y_2)
\end{equation*}
Lexicographic preferences have two important properties:
- $x \sim y$ if and only if $x=y$
- they have no utility representation
If $x=(1,1)$, $y=(0,1)$ and $z=(0,0)$, we have $x \succ y \succ z$. In addition, any $\lambda>0$ is such that $\lambda x + (1-\lambda) z \succ y$. Thus assumption 2. is not valid, and the conclusion of the theorem (existence of a utility representation) is not valid either.