Many log flumes do have brakes. You can see them, they push against the sides of the boat and hold it against the current.
Most of the ones I've seen push up from the bottom. I have see the side-gates, though.
Many log flumes are gravity fed. The lift hill is "dry" and there's a pipe that goes to the top to have water cascade down the drop side. That's mostly for show, though. You're going down the hill on the wheels on the bottom of your boat and tracks just to the side instead of down some water plume. If you look, there's typically a basin underneath the drop hill where a lot of water is being pumped in (huge pipe and pump). This fills up the splash zone where you touchdown which, again, is mostly done with the wheel and tracks up to the very end of the splash. It keeps you straight and safe in the process. At that point, traditional log flumes will be gravity based until the next lift hill or station. There may be some jets in flumes like Splash Mountain to keep you going.
With regards to safety:
- There are brakes in the station and sometimes they use gates. I think Splash Mountain uses gates, if I remember.
- They have the rubber conveyor belts there that lift you out of the water at Splash Mountain and provide a stable surface for loading and unloading whereas traditional log flumes would have the underneath-brake which is a set of wood/metal planks with rubber padding. They push those up enough to where it either catches the boat or stops it and you load/unload.
- Other than that, as mentioned before, if you need to emergency stop then you just stop the lift hill and stop any action in the loading/unloading area. Everyone else just floats forward until they bump into the log in front of them.
- You want to keep the trough full, though, as you can have a nasty accident if a log is coming down the drop and there's no water there to slow it down and float it at the end of the aforementioned track. There was a case like this at Six Flags Over Georgia, years back, that really hurt someone and resulted in a lawsuit.
- If you look at the logs, themselves, then you'll see that they have wheels on the bottom of the boat (and I've never been a fan of partially empty troughs where you're just rolling on the bottom) and along the sides of the boat, but still on the bottom, to keep you floating along in the trough.
Other than that it's pretty much: Guests stay in the boats unless told to by employees. There's not much else to them. I think the water is about knee deep in most of them so you'd have to be trying to drown. A bigger issue would be if you got out of the boat and then were in the path of the other boats. The boats themselves are pretty heavy and if you add a fully loaded boat then that's a lot of mass coming at you that is going to use you to stop. I bet you'd be seriously injured or killed in that situation. I've never seen anyone get out of a boat on a log flume nor heard of it. Overall they're pretty safe rides. You really just need the pumps working.
With that said, you're probably thinking of a power outage and what happens if that pump isn't filling the trough that you land in from the drop: It's true that it will drain out but it won't be immediate so if someone was at the top of the drop and coming down and there was a power failure just as they crested, there'd still be plenty of water for them at the bottom. I'd also guess that there's some kind of mechanism or procedure in such events so that when the power comes back on it waits for the trough to fill before starting the lift hill back up.
