It is possible to find such metrics. However, they are a bit unnatural, in that they aren't really constructed from any inherent properties of matrices. For concreteness, I'm going to restrict my attention to $M_2$ and $Gl_2$, but this construction can be modified to work for any larger $n$ as well.
We consider $2 \times 2$ matrices $$
\left[
\begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right]. $$
To construct the desired metric, we start by changing coordinates on $M_2$ as follows.
$x = a+d$, $y= \exp(a-d)$, $z=b$, $w=c$
This gives a map from $M_2$ to a half-space in $\mathbb{R}^4$. For a Riemannian metric, we use $$ds^2 = \frac{dx^2+dy^2}{y^2} +dz^2+dw^2.$$
Notice that the $x$-coordinate is simply the trace of the original matrix, so what's left to show is that $x$ is bounded on any geodesic. Different geodesics will have different bounds, however.
To show that $x$ is bounded on any geodesic, observe that as a Riemannian manifold, $(M_n, g)$ is simply the metric product $\mathbb{H} \times \mathbb{R}^2$. As such, the geodesics are of the form $\gamma(t) = (c(t), \ell(t))$.
Here, $c(t)$ forms an arc of a semi-circle in $\mathbb{H} $ whose center is on the axis $y=0$. Meanwhile, $\ell(t))$ is a line in $\mathbb{R}^2$. As such, it follows the $x$ coordinate is bounded by the center and radius of $c(t)$.
When we try to do this for $Gl_2$, we have to deal with the fact that the space is disconnected. However, we can use a similar idea on each connected component. Namely, we change coordinates to
$x = a+d$, $y= \exp(a-d)$, $z=b$, $w=ad-bc$ and take the metric
$ds^2 = \frac{dx^2+dy^2}{y^2} +\frac{dz^2+dw^2}{w^2}.$
This gives $Gl_2$ the metric structure of $\mathbb{H} \times \mathbb{H}$ on each of its connected components, which also has the desired geodesic property.
