Ok, I think I've solved the mystery.
I went back and tried to play the first link from the first post for myself this time, using VLC. It only played for 41 seconds then stopped.
I then tried playing the stream using the link from the web site,
http://www.latvdefrance.com/stream/live/name/tf1. It played from 01:44:38 to 01:45:19 (the last 41 seconds) then stopped.
When I first visited the web site it said, "Vous n’êtes pas connecté, votre accès est limité à une minute! Créez un compte pour accéder à 12h d’essai gratuit".(Translation: You are not logged in, your access is limited to one minute! Create an account to access 12 hours of free trial). Even after I created an account and logged in, it still wouldn't play more than the last 41 seconds of the stream. There were also messages about requiring a paid subscription to view the live streams.
I tried playing their streams from previous days' TV on my Roku. They played just fine.
Conclusion: The live streams won't play more than a minute's worth of content unless you supply the appropriate credentials for a valid paid-up subscriber account.
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It still seems a little odd that the server would generate a playlist file reusing the same 6 segment files over and over again. I'm not surprised though that VLC and my browser's media player only played the last 41 seconds, whereas the Roku kept looping, It's been my observation that the looping is normal behavior for the Roku when it gets confused playing an HLS stream.