Disclaimer: I'm not advising anyone with this discussion - just discussing what I've done and what it might mean.
With that disclosure, I'm not really sure where stitches might come in on this. I can see it possibly cracking or breaking the edges. However, the process I use with the hand corker seems like quite a gentle process. There is no jerking, and surprisingly (when I first used the corker) is that the constricting spring doesn't really need the lip to work. It's really just gentle pressure on the top, and I use a slow and gentle downward motion. When I first tried, I noticed that the plunger didn't center on the corks, but if I started with a little downward pressure, then let the pressure off just a bit, the cork would adjust and the plunger would center when I began the downward pressure again.
It goes in pretty smoothly. I suppose it is possible to break the upper edge of the lip with the downward pressure (since the constrictor spring doesn't really hold it in place, it must be the downward pressure I'm applying), but again, the pressure used is pretty gentle. There is also pressure on the lip when using the corkscrew to remove the cork.
There is no denying that the screw-tops are thinner at the very top than the other wine/champagne bottles, so there would be a greater hazard for breaking the lip than with the normal wine bottles. They don't seem to be any thinner after you get past the threads. Defining how great the hazard is a good discussion. There's little reason to actually do it as is pointed out, but how big is the actual hazard? hmm ... I'll leave it up to you to decide.
Repeating my disclaimer: I'm not advising anyone with this post, but just discussing what I did and what I think as doing it.