Show there is an uncut square lying in a larger square cut by lines

Let the large square have side length $3$. Note that in the figures of the OP appear two completely different limiting cases where no small square with side length $>1$ can be placed. Any proof will have to incorporate these cases somehow.

If there are no cutting lines we can move a small (unit) square along the inner boundary of the large square along a track of total length $8$. We shall show that a single cutting line $\ell$ leaves a part of length $\geq4$ of this track available for the small square. It follows that two cutting lines cannot make all points of the track unavailable.

Place the large square with its center at the origin. By symmetry, it is enough to consider cutting lines $\ell$ given by an equation of the form $y=\tan\alpha \cdot x+c$ with $0\leq\alpha\leq{\pi\over4}$ and $c\geq0$. Such a line $\ell$ will cut $n\in\{0,1,2\}$ little corner squares (shaded grey in the above figures). The case $n=0$ is not drawn; the upper two figures show the case $n=1$, and the lower two figures show the case $n=2$. It is easily verified that in each case the pink squares can move freely along a total track length $\geq4$.