The mobile half of Catch-Up Scheduler: an iOS Shortcut finds what's new, this page plays it back to back. No account, no server, no data leaving your phone.
Two Shortcuts, two different jobs:
The one that actually runs. Checks your channels for new uploads and plays them, on the schedule you set in step 4. Works immediately with a small built-in example list — nothing else required.
Only needed if you want your own channels instead of the built-in example. Point it at a .catchup file and it becomes the active list.
Build a list of your own channels, save the file, then run Load Catch-Up once to switch to it.
Open it once in Safari, then Share → Add to Home Screen, so it opens full-screen like an app whenever the Shortcut hands it a queue.
Shortcuts app → Automation → Time of Day → your usual free time → run "Catch-Up Check" → turn off "Ask Before Running". Or just tap it manually whenever you're free.