This seems to have become popular in the jargon of business and politics where it means a point at which there is intense tension and conflict. Example sentences are: The employers have reached a pinch point with the unions. Nauru is a pinch point for the electorate.
The term has been around in other more specialised areas of language. In maths it is the point where a surface touches itself tangentially. It is also used in relation to machinery with moving parts as the point where the human operator is most at risk of being caught by the machine. It is used in relation to traffic where it means a point of greatest congestion.
In its new metaphorical use it is a synonym for crisis – but with a much greater ouch factor than crisis. We can feel the pain.