“Options are rarely exercised” – why exactly does the *final* holder bother buying them, in that case?

I'm very much new to the whole world of options, calls, puts, etc. I have an understanding of what the various terms mean and how they work, but my interest in options is very much limited to how options activity by other people can sometimes impact the share price itself through run-ups, etc. So I'm not really asking this with a view to trading options myself, more just for general information about how everything works.

I can't wrap my head around the basic, apparently universal concept that “options are rarely exercised”. The premise of this saying is that the majority of options trading involves buying and selling options as opposed to exercising them for the underlying stocks – in other words, people who trade options are far more likely to be looking for profits in terms of how much they can re-sell an option for after buying it, hopefully re-selling it to someone else at a much higher price.

What I don't understand about this is, surely somebody must eventually exercise ITM options, otherwise they would be entirely worthless right from the get-go? Hypothetically speaking if an option is going to expire ITM this coming Friday, and it's just being bought and re-sold again and again by different people as the value increases, at some point on Friday, somebody is going to be the last person holding it and faced with the choice of either exercising it, or having it expire worthless?

In that context, how is it possible that most options are never exercised at all? Does that essentially mean that your average options contract is a bizarre game of pass the parcel with somebody inevitably left holding the unopened parcel at the end of the game?

If the rule of thumb was “most traders don't exercise options and instead make money on the trading thereof”, that would make perfect sense. But the idea that most options themselves are left to expire unexercised while people have still made money buying and selling them seems very strange to me, as I don't understand why anyone would buy an option that's on the brink of expiry if they had no plans to exercise, but not enough time to sell it on to someone else.

Am I missing something here? Or is the phrase “options are rarely exercised” an inaccurate way of saying “options traders rarely exercise”?