Since the question is related to making the Poisson distribution more understandable, I'll give it a go, since I recently looked into this somewhat for call center incoming call patterns (which follow a memory-less, exponential distribution as time goes on).
I think delving into another tangential model that essentially requires knowledge of Poisson to realize how it isn't one may be somewhat confusing, but that's just me.
I think the trouble with understanding Poisson is the continuous time axis it's on --- as every second goes on, the event is no more likely to occur --- but the further out in the future you go, the more certain it is of happening.
Really, I think it simplifies the understanding if you just trade the 'time' axis for 'trials' or 'events'.
Someone can correct me if this is way off base, as I feel it's an easy explanation, but I think you can replace the flip of a coin, or the toss of a dice, with 'time until a phone call arrives' (what I typically use for Erlang C/ call center staffing).
Instead of 'time until a phone calls arrive' ---- you can replace it with ... 'rolls until a dice hits six'.
That follows the same general logic. The probability (like any gambling) is completely independent every roll (or minute) and is memory-less. However, the likelihood of 'no 6' decreases ever slower but surely towards 0 as you increase number of trials. It's easier if you see both graphs (likelihood of call with time, vs. likelihood of six with rolls).
I don't know if that makes sense --- that's what helped me put it together into concrete terms. Now, the poisson distribution is a count rather than 'time between calls' or 'trials until rolling a six' -- but it relies on this likelihood.