Movies just stop, but TV shows will flow through to the next episode within a season on many Netflix platforms (including Chromecast).

It's really never made sense to me. Sure, play two in a row, but why would Netflix keep playing episodes any incur the royalty costs when people fall asleep watching a show, don't turn off the service when they turn off the TV, just plain walk out with the show playing, kids shows keep playing with no one watching?

I assume they incur royalty costs, or is there some incentive for them to somehow boost the number of hours of content played? Is this one of those cases where they are trying to inflate their numbers to look good to investors?

It will play 2-3 in a row without any user interaction, but if you haven't taken any action in a while it'll stop and give a modal "Continue Playing?" message

It does, at least on TV series it autoplays the next episode. Definitely does that on the Apple TV.

After a few (3?) episodes, Apple TV pauses the stream and asks if you are still watching.

My Roku does this after three episodes as well, although that adds up to quite a few megabits over a three-hour period.