You can do exactly that - set up an Assist pipeline that glues together services running wherever you want, including a GPU node for faster-whisper. The HA interface even has a screen where you can test your pipeline with your computer’s microphone.

It’s not exactly batteries-included, and doesn’t exercise the on-device wake word detection that satellite hardware would provide, but it’s doable.

But I don’t know that the unit will be an “upgrade” over most headsets. These devices are designed to be cheap, low-power, and have to function in tougher scenarios than speaking directly into a boom mic.