Maybe a DLNA server like miniDLNA might be what you are looking for. It does not have a website as an interface, but you are able to stream from it using VLC.
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You can use ffmpeg to get the M4A and stream that instead of the whole MP4, could probably automate it too if you want to stream an MP4 that hasn't been processed yet.
This one I am not sure about, but maybe SubSonic Advanced might be able to do it. I know video support is mentioned but I don't know if that is video and audio or just the audio of the video.
https://github.com/airsonic-advanced/airsonic-advanced.
Note, it has been almost 3 years since the last update.
I've not used it personally (I use Plex), but I have heard great things about Jellyfin. Does it maybe have the feature you need?
Jellyfin.org
I prefer Jellyfin over Plex, but it doesn't look like it has this feature, yet (https://features.jellyfin.org/posts/533/audio-only-option-on-video-playback)
Are they accepting pull requests?