Does taking live photos store a redundant MOV file?
It is not redundant. Live Photos consist of an high-res image and a video (lower quality than the image).

The HEIC (High Efficiency Image Codec) format is only for images, and it's the one when you actually hit the exposure button.
The .MOV files are displayed when you long press the photo.

Therefore you can extract the single HEIC image from the MOV file (with lower quality though), while you can't turn the single image into a video. This also means, this is something the iPhone is doing and not photo sync. You would also get those files when using AirDrop for example.

Note: Since the video is lower quality than a single image, changing the key frame must change the quality. However, I could imagine (this is speculation though) that Apple might use some ML/smart algorithm to make the new key frame look better (similar to something like night mode, where a lot of processing happens).

Sources:

How does changing key photo in Live photo differ from burst mode?
Live Photos Called 'Best New Feature,' Composed of Separate JPG and MOV Files
Apple Live Photo file format on SO
Live Photo Quality Loss When Changing Key Frame on Apple forums