With jazz, anything goes. That said, I don't see much use in playing a cornet that inherently sounds like, or is made to sound like (by mouthpiece selection and/or playing style) a trumpet. I learned to play on an American wrap cornet back in the 1960's, and then "stepped up" to a trumpet in high school. It took me until the early 1980's to gain an appreciation for cornets and begin to learn the nuances in playing them. Today, I play cornet more than trumpet, but both instruments have their place. I'll drag out my trumpet and play it when I need to, and I've doubled those two instruments so long that I think I have a pretty good handle on playing each with the necessary tone and style that's expected.
The videos posted by the OP do sound like a cornet...jazz style. It's not the traditional brass band cornet sound or style (which I think is the historically correct sound of a cornet), though. Most of the videos I've seen of folks playing jazz cornet sound like a trumpet. If I closed my eyes, I wouldn't be able to tell which instrument was being played.