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More generally, regarding "a better way": For any matrices $A$ and $B$, let $F(x,y,z) = \det(z \mathrm{Id} + x A + y B)$. This defines a curve in $\mathbb{P}^2$. Your condition is every line through $(0:0:1)$ intersects this curve in a point of multiplicity $\geq m$. This turns out to imply that $F$ must have a component of multiplicity $\geq m$: i.e. $F$ factors as $G^m H$ for some homogenous polynomials $G$ and $H$.

If $A$ and $B$ had a common eigenspace, then $G$ would be linear, and $A$ and $B$ would have direct sum decompositions $A = A_1^{\oplus m} \oplus A_2$ and $B = B_1^{\oplus m} \oplus B_2$ where $(A_1, B_1)$ and $(A_2, B_2)$ give rise to the plane curves $G$ and $H$.

The preceeding examples show that life can be more complicated. In the first example, $F=G^2$, for some nonlinear $G$. In the second, the factorization of $F$ does not correspond to a direct sum decomposition of $(A,B)$.

There is a fair amount of literature on expressing plane curves as determinants, so there is probably someone who has looked specifically in the case of a multiple factor, but I don't know who.

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This is false. Let $A_0$ and $B_0$ be $k \times k$ symmetric matrices with no common eigenspace. Let $A$ and $B$ be the $2k \times 2k$ matrices with block forms $\left( \begin{smallmatrix} A_0 & 0 \\ 0 & A_0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} B_0 & 0 \\ 0 & B_0 \end{smallmatrix} \right)$. Then $A$ and $B$ have no common eigenspace, but every eigenvalue of $Ax+By$ has multiplicity $\geq 2$. And, of course, you can replace $2$ by any $m$.

Here is a different example. Let $A$ and $B$ be symmetric matrices of the form $$\begin{pmatrix} \ast & \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast & \ast & \ast \\ \ast & \ast & 0 & 0 & 0 & 0 \\ \ast & \ast & 0 & 0 & 0 & 0 \\ \ast & \ast & 0 & 0 & 0 & 0 \\ \ast & \ast & 0 & 0 & 0 & 0 \\ \end{pmatrix}$$

Then $Ax+By$ is also of this form, and you can check that any matrix of this form has a zero eigenspace of multiplicity $\geq 2$. However, there is no reason for the $0$-eigenspaces of $A$ and $B$ to meet at all!