How to get the "real" stream time for the most recently switched-to game for a stream?

Other statistics people don’t have a problem with this…

Make one!

Sounds you you seem to be collecting data for no particular user. Best make sure whatever you are doing abides by the Developer TOS that you already agreed to when you created a Twitch ClientID.

Considering further more you are hiding behind a fake name.

https://www.twitch.tv/p/en-gb/legal/developer-agreement/

Specifically around data collection and storage. Additionally with GDPR when collecting data in the EU or for users from the EU

It’s not missing. It’s in the streams end point if you continually poll for data changes, including but not limited to, category, stream title, tags, or viewer count