I've just checked two of my S&Ws with the lock and the flags don't move on their own when I push on the cylinder latches.
I then used a probe to pull up the flag and there was just a tiny amount of spring pressure which brought the flag back down if pulled up.
I'm not well-versed enough to disassemble the lock-work in my external hammer S&W's any further at this late hour as I've got to be somewhere for work in the morning. Also, I've never done a full detail strip and there's always that possibility I'd never get it back together.
Perhaps someone else with more knowledge of that lock with the indicator flag will answer soon who can definitely diagnose your issue.
I'm suspecting that flag is either broken or not perfectly assembled to provide that spring tension to keep that flag down until the key is actually turned to bring it up.
I'm curious, too, yet glad my guns aren't doing what yours is doing as it doesn't seem right. I'll go look at some Youtube videos next to see if somebody is shown completely detail stripping that lock area. I'll subscribe to this thread for my own edification.
OK, go to ~5:09 on this Youtube vid and see how the little spring on the flag should be placed. I'm guessing either that spring is broken or just out of place. You'll need to dissemble to the point of getting the bolt out of the way to access the flag & spring.
The vid is of someone working on a 640, which means there are a few less parts compared to an external hammer model, which makes things slightly easier for that guy.