Your intuition can be stated as the proposition below:
If $A$ and $B$ are two different points on a hyperbola and the midpoint of $AB$ lies on the asymptotes, then this midpoint is the center of the hyperbola.
Proof: Note that this proposition is irrelevant to coordinate systems, thus without loss of generality, the hyperbola can be assumed to be $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$.
Suppose the coordinates of $A$ and $B$ are $(x_1, y_1)$ and $(x_2, y_2)$, respectively, then $x_1, x_2 ≠ 0$ and$$
\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = 1, \quad \frac{x_2^2}{a^2} - \frac{y_2^2}{b^2} = 1. \tag{1}
$$
Because the asymptotes are $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 0$ and $\left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)$ lies on the asymptotes, then$$
\frac{(x_1 + x_2)^2}{a^2} - \frac{(y_1 + y_2)^2}{b^2} = 0. \tag{2}
$$
Now, suppose $x_1 + x_2 ≠ 0$, then $y_1 + y_2 ≠ 0$ by (2). From (1) there is$$
\frac{x_1^2 - x_2^2}{a^2} = \frac{y_1^2 - y_2^2}{b^2},
$$
and from (2) there is$$
\frac{(x_1 + x_2)^2}{a^2} = \frac{(y_1 + y_2)^2}{b^2},
$$
thus\begin{align*}
&\mathrel{\phantom{\Longrightarrow}}{} \frac{x_1 - x_2}{x_1 + x_2} = \frac{\dfrac{x_1^2 - x_2^2}{a^2}}{\dfrac{(x_1 + x_2)^2}{a^2}} = \frac{\dfrac{y_1^2 - y_2^2}{b^2}}{\dfrac{(y_1 + y_2)^2}{b^2}} = \frac{y_1 - y_2}{y_1 + y_2}\\
&\Longrightarrow \frac{2x_1}{x_1 + x_2} = \frac{x_1 - x_2}{x_1 + x_2} + 1 = \frac{y_1 - y_2}{y_1 + y_2} + 1 = \frac{2y_1}{y_1 + y_2}\\
&\Longrightarrow \frac{y_1}{x_1} = \frac{y_1 + y_2}{x_1 + x_2} \Longrightarrow \frac{y_1}{x_1} = \frac{y_2}{x_2} := c.
\end{align*}
Note that $x_1 + x_2 ≠ 0$, plugging $y_1 = cx_1$ and $y_2 = cx_2$ into (2) to get $\dfrac{1}{a^2} - \dfrac{c^2}{b^2} = 0$, then by (1) there is$$
1 = \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = \left( \frac{1}{a^2} - \frac{c^2}{b^2} \right) x_1^2 = 0,
$$
a contradiction. Therefore, $x_1 + x_2 = 0$, which implies $y_1 + y_2 = 0$ by (2). Hence the midpoint of $AB$ is the center $(0, 0)$ of the hyperbola.
