Which streaming service? I last used yt-dlp with the --live-from-start option on Youtube, it worked fine there.

You didn’t specify a streaming platform. Assuming you’re referring to twitch vods I would actually recommend using twitchpipe instead of something like yt-dlp as another user recommended. Reason being is that twitchpipe comes with a shell script that lets twitchpipe monitor a twitch channel for when it goes live, automatically starts recording the vod, and stops when the stream ends - which I find more ideal for putting it on a vps and letting it do its thing.

Depends if the stream is clear or protected with some kind of DRM. I would say you can try with ffmpeg. Just open the network tab in developers tools and look for a GET request that downloads some file ending with .mpd (MPEG-Dash) or .mp3u8 (HLS) and provide this URL to ffmpeg as an input.

You can save files via streamlink: https://streamlink.github.io/