For live streaming, object storage won't do you much. That's useful for VODs I guess but the live stream there ain't much you can do.
You can use a cheap VPS as a proxy to your home server for the IP hiding aspect. Oracle's free tier might do okay for that purpose, just make sure you keep local backups. Or just Cloudflare Tunnels. A VPS is nice though because you can upload once to the VPS, and it'll redistribute to the viewers and will have much more bandwidth available.
But self hosting, especially video, generally ain't cheap. That's why the big guys like Twitch and YouTube are so invasive with ads and subscriptions that subsidize all the free users.