Fluoroalkyl chemicals are only "inert and unreactive" in a relatively narrow sense of "wouldn't catch fire", "don't react with strong acids and bases", and similar.

They are plenty reactive in a sense of interacting with enzymes and other cellular machinery.

Not really accurate. These chemicals are quite unreactive. Precursors from manufacturing waste can be very reactive, but most of the problematic contamination regards the forever chemicals themselves, not precursors. This paper is probably the best scientific review of what is going on in the human body. https://www.sciencedirect.com/science/article/abs/pii/S03043...

Maybe sci-hub has a copy of the full paper. Not sure.

As briefly as possible, and therefore glossing over many many details, the toxic effects are mainly due to cell membrane perturbation, cell membrane transport disruption, and binding to hydrophobic protein cavities (thus disrupting the usual function of these cavities).