Conductively, I'm not so sure. Assuming this new-fangled connector is well designed (to ensure good contact with all the inserted wires), I think I could easily be convinced that the overall connection is electrically more reliable than just twisting several wires of any serious gauge together... And as for "mechanical strength", do I even want to know in what application you're relying on wires twisted together with a nut for any serious "strength" to begin with?Honest question, do you not twist your wires together before you install your wirenut? There's no way that push-ins should ever be stronger mechanically or conductively than a proper wirenut connection.
A good wire nut, properly sized for the amount of wires going into it, can be a bit better. But it does depend on pretwisting the wires, which can be hard to do when you get to 3 or 4 of them. The bests wire nuts have the little ears to make it easier to get them nice and tight. There is also a slim chance (and I have seen it happen) that a wire will pull out of a wire nut. This can not happen with the item discussed in this post.