No they don't. The O-ring can get pinched in the hinge area during flex but the rubber is pretty soft and it squishes quite easily, and that usually only happens if you are trying to push the functional angle limits of the flex joint. If the O-ring is pinched too hard it will sometimes pop out of position on one or two corners of the flex joint and you will just have to flip it back into the flex joint channel when you are done using the socket.
The O-ring does not hold everything super rigid, it holds a flex socket so that even with zero internal hinge pin resistance the socket can be angled into any position by hand and when you let go it will stay there on its' own. That's what most people need in order to feed a flex socket into a tight area and place it on a bolt head without any secondary assistance or hassle.
With a socket on a universal joint the hinge usually won't stay fixed in place (due to the weight and leverage of the socket hanging off the end) but the O-ring will still give you much more control over how the socket is oriented than you had originally. If you occasionally need a more rigid connection when using a universal joint you can still use the trick of wrapping the joint with electrical tape as a secondary aid. Each wrap of the tape makes the joint a little bit more rigid while still allowing the socket to rotate and flex during use.