This often happened, though not always.
An important counterexample are 3D graphics cards, which basically put the OpenGL/DirectX fixed-function pipeline into silicon. It took a long time and many iterations to make the pipeline more programmable until the 3D graphics cards turned into modern GPUs.
Even today, GPUs live on as separate hardware in a computer instead of having become integrated into, say, the CPU. Intel's attempt to do something like this with the Larrabee project [1] was discontinued.
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[1] https://en.wikipedia.org/w/index.php?title=Larrabee_(microar...