I will not speak to the code issue as other already have. But, I did use them when building my new home a few years ago and, for me, they passed inspection.
While the package does say they can be used for stranded wire, I did not find that they worked that easily with stranded wire (I was using some computer controlled switches with pigtails instead of screws). But, they did work. I just had to twist the strands fairly tight first.
As far as coming loose? Not the ones I was using. If memory serves me correctly, I think there is a little hole that you can push the spring up with to release the wire because it is so tight.
Any wire connector can fail. Screws can loosen with vibration, stress, corrosion, temperature fluctuations, and so on. Push-in (or stab-in) connections can fail for the same reasons. Crimp and pin connections, friction lock, and even soldered joints can fail.
The real question is: what's the most effective wire connection given the operating conditions, parts and installation cost, lifetime expectancy, use case, and so on.
For example, where I work (we manufacture large electric signs), we use all manner of connection methods, and they vary based on these factors. One such factor is serviceability. If a connection is expected to be rugged and reliable but disconnected for safety reasons, we may use screw terminal blocks (which only accept straight wire) or friction lock (which requires use of a screwdriver to release, similar to the push-in connectors on outlets and switches).
Outlets shouldn't require frequent service nor replacement, nor should experience extreme vibration. They also won't (usually) have wide temperature swings, which leaves the stress from normal usage (connecting and disconnecting plugs) which can wiggle things loose over time.
A well-made outlet can have quite good push-in connectors; by "good" I mean they hold solidly for years and aren't likely to fail with normal use. However it can be difficult to tell during installation whether the inserted wire has made good contact or not. Also if one happens to be faulty, it may accept the wire but provide little feedback that the locking mechanism is less than optimal. Pulling on the wire after installation is one way to tell, but if you're using the push-in connectors, are you spending the extra time to do the pull test? Probably not. Screw terminals are easy to visually inspect and determine if solid contact is being made and that the wire is firmly held by the screw or plate.
So while push-in or friction lock connectors can be quick and useful in certain applications (with appropriate confirmation of operation), I recommend wrapping wire around the screw terminals for the most durable connection that's easily verifiable. Usage and vibration can still loosen the screw, and in such case the wire hook will be less likely to separate from the outlet than a straight wire. In all cases where the screw comes loose, however, arc fault can occur, causing problems.
Best Answer
I use one of the flat blade screwdrivers from this tool set from General. This is available from many better hardware stores or directly from the General Web Site.
I have found that the flat ends on these are formed much longer and the poke down into the slot on the back of the outlet much better. The flats on the type of screw driver bits that you pictured generally get thicker within a much shorter length.
When you replace the outlets / switches with new ones do use the screws instead of the poke in wire holes. This will produce much more reliable connections. Another thing to watch for is to place the wire hook under the screw head so that the end of the hook goes around the screw in the same direction that you use to tighten the screws.