Why are LTI filters not able to output frequencies not present at the input?
Because for an LTI system, every input sine has an output which is a sine of the same frequency, with a potential shift and an amplification factor.  In other words, input $x(t) = a\sin(\omega t+c)$ will be turned into some $y(t) = aH_\omega \sin(\omega t+d)$, where $H_\omega$ is the amplification factor for every sine  with frequency $\omega $.

This come from the theory: LTI are diagonalized by complex exponentials (in a pedantic way). As a result, an absent sine at a given frequency is a zero amplitude sine (in other words, $a=0$), and the output will be a shifted sine, with   zero amplitude whatever the amplification $H_\omega$  (since $aH_\omega=0$), still a zero amplitude sine.

More details:


Are complex exponentials the only eigenfunctions of LTI systems
Impulse Response to LTI