I doubt that'll happen, and I doubt this'll be the last chance to catch Pinsir. The Rotoms have already come back so I'm sure Pinsir will as well if people fail to catch it now.
Nintendo understand that the point of Pokémon is to "catch 'em all," and they're not going to make it impossible for someone starting the game next Tuesday to do that.
It's exactly how all free-to-play games work. You can play it and just accept the pace you unlock stuff at, or you can WIN EVERYTHING RIGHT NOW by using your in-game currency. Both of them, ultimately, are going to lead you to the same goal.
I'm sure of someone posted "Give it a few weeks, you'll only have 4 chances to catch a pokemon with a very low catch % then you can't battle him again whether you catch him or not" you probably would have said "I doubt that'll happen" back then too. We also don't know how they are doing returning pokemon or which pokemon are even returning, this week is really the first time a pokemon has returned (outside of repeated weeklys which I think don't count) and the game has been out for several months now. How long until people see Pinsir again? 2016? 2017? Never? Not really something I expect most people to commit to in the hopes of getting a repeat Pinsir stage.
When things like this start happening, it's only the beginning. It gets much worse and gradually forces you into accepting paying for every pokemon stage, because by that point it's no longer "meh i'm quitting, it's pay to win" but rather to "meh, I had to spend every 2nd pokemon anyway, may aswell pay for this one aswell". Just look at how the game used to be even just a month ago to how it is these past few weeks, definitely a huge change.
Meh, if I get get to the point where I say to myself "To get this pokemon I need to spend real life money, should I or should I not?" I'll quit right there and then. Right now it's not so bad as I've said so I'll continue playing, but at this rate I'm not sure how much longer I'll be playing for.