Photo stream is not the same as iCloud Photo library (which puts your full library from the Photos app in the cloud). The latter requires a 4th gen or newer Apple TV.
Photo stream is for automatically syncing photos taken on IOS across devices, if photo stream is enabled on your Mac then photos imported there will also be a part of the photo stream. This is not meant as a storage solution, again these photos are only there for 30 days.
If you are using home sharing then that is separate and done via the Computers app on Apple TV (in this case your computer does need to be on and have iTunes running as it is accessing your local content).
If you are using photos for a screensaver, whether from the cloud or your computer, then only a select amount will be temporarily cached. If you want to see different photos then you need to manually go into the screensaver settings and select a new batch to be cached.