How do I cope when my game's story keeps getting cut?

You can build a story a player lives inside rather than merely watches. That is a deep craft. And it hurts that story is so often the first thing dropped when time or money runs short. You pour yourself into something that gets trimmed away, and calling that grief is not too strong a word. It helps to see why it happens. Story is the layer players cannot point to on a feature list, so it looks cuttable to people counting deadlines. That does not mean it does not matter. The best-remembered games are remembered for how they made people feel, and that feeling is your work, even when it survives in fragments. Protect what you can by weaving story into the systems that will not get cut, so it cannot be cleanly removed. And let yourself mourn what is lost without letting it turn you bitter. You are making something that lingers in people long after they stop playing. That is worth the ache of the parts that never made it.