FreeTube does not have controller support, and for AndroidTV I'd recommend SmartTube.
Kodi/LibreELEC is able to do all of it, but IMO it's not a good experience for browsing YouTube and I don't know how well the third party Steam Link integrations work.
This is why I'd also recommend LineageOS Android TV, which supports Pi's thanks to konstakang. But I'm not sure why it'd work better than a FireTV stick, since both run AndroidTV.
Edit: I've had an issue where the Pi 5 wouldn't boot AndroidTV, until I tried to turn it on again after a few weeks. So I'd recommend sticking with the FireTV + SmartTube + Jellyfin + Steam Link (unless you've got a Pi 5 lying around anyway).
Edit 2: The Pi 5 + Android TV had issues with HDMI-CEC of the TV, so I had to buy a remote with a USB adapter. This sends the wrong signals (e.g. keyboard enter, not what Android TV expects), which is fixable with some app remapper. Maybe it'll work better for you, but the FireTV is likely the easier solution.