I have both the Escoulen chucks and have looked at the Axminster Eccentric chuck. The latest Escoulen chuck will do what the Axminster chuck will do which is off center turning. You are correct in that the axis remains parallel, but what you can do is have several profiles at various points or when doing the inside of a bowl, have several "bowls". Both chucks will do this.
The Escoulen chuck allows you to do "eccentric" turning, in that you can angle or "cant" the work from the axis. I am working on some bowls at this time which are both off center and eccentric. The center of the blank starts out off center and the piece canted at 8 degrees. Depending on the amount of off center and which direction, you may wind up with base that will remain in the center of the blank or off to one side. I have done both. I have one bowl that has two different profiles on the outside (180 degrees apart) and two intersecting profiles on the inside, with part of the rim at another angle. And the foot is somewhere around the center, also having two different profiles.
The resulting bowls are not for everyone, one of those personal taste items. The Escoulen chuck is also not for everyone, it does take a substantial lathe. Last night I had a small 6" blank mounted and was shaking a 1000lb lathe until I removed a lot of mass. But, turning the elliptical rims are not much worse that square or natural edge bowls. You just have to becareful when sanding.
There is also the Sorby offcenter chuck, which is a little easier than the ones I have. I wish that the Escoulen chuck had the Sorby indexing mechanism. I can do the same thing with mine, but it is just a lot more difficult. But, any of them will let you explore some of those way of adding a little touch that makes people wonder how you did that. For me well worth the time and keeps the items from getting routine. With the current sale at CS, it would be a good way to find out if you like these forms or not...