Low-pass vs median filter delay
i would say that your reasoning is a teeny bit flawed.

the delay of an LTI filter is a deterministic function (of frequency) that is unambiguous.  a sinusoid of some known frequency goes in and the sinusoid that comes out will be delayed by a predetermined amount of time.

a median filter is not linear.  and, if a sinusoid goes in, it will not be a sinusoid coming out.  let's say that your causal median filter is outputting the median value of the current input sample and the most recent past $N-1$ samples (the "formulation" may seem "simple", but you're gonna need to repeatedly sort $N$ samples to pick out your median value), you can come up with an expected value of the delay and, since the sample that is eventually chosen to be the median value can be any one those $N$ samples with equal likelihood, then the expected value of the delay is $\frac{N-1}{2}$.