It's absolutely a cost-saving decision, no doubt in my mind.
Putting my day-job hat on, they've surely got a static media source that you can seek back and forth on that's actually used as the "live" feed, as otherwise you'd have to look at what you suggested around building buffers per session if somebody paused which would get expensive quickly.
Regardless, the UX is poor and it needs to configurable, if not removed entirely.
