Why are fields with characteristic 2 so pathological?

Mostly, it's because $a=-a$ in fields of characteristic $2$.

All fields of nonzero characteristic are 'pathological' in some sense. It's just easier to trip over a problem with $2$ than a problem with, say, $1319$.

Symmetric nilpotents exist in all characteristics. For example, in characteristic 3, you have

$$ \left( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right) $$

as an example of a matrix that squares to zero. It's easy to generalize this to any positive characteristic.

Squaring behaves strangely in characteristic 2. Among the oddities is that there is only one square root of 1. In some sense, this is responsible for the thing with symmetric and anti-symmetric.

In characteristic 3, it's cubing that's strange, and so forth.