Zoom has actually worked pretty well for the LIVE stream component and some of my athletes and S&C’s have used it to run spin classes and circuits. You can record the session…. but that is probably where the functionality ends in terms of being able to automate it. That said, the cost is right (being free). The sticking point will be if your sessions last for than 40min with more than 3 users… then you will need a subscription. The good thing is by recording the screen, you get the video and audio file to cut up and play with.
I have tried a lot of exercise prescription software and the one I believe has the most flexibility around uploading, tagging and filtering your own content is Visual Coaching Pro. They are based in Melbourne. Their standard library is ~6500 exercises, but more recently they have allowed users to record their own exercises. Its not LIVE though. You would use the recorded exercises to build your session, which actually could be a plus moving forward, as it becomes easier to keep your content fresh, reuse content or add more. Their system is easily placed inside an i-frame on a 3rd party membership site… could be an option for you. If you talk to them, ask for Luke.
Take care Jamie