Paddlefish aren't kept because of several factors.
First, they need a round tank. A squared off tank will cause them to damage their bill which can be fatal over time.
Second, unless they're raised from fry on pellets, they will only feed on plankton. Once they're large, they're nearly impossible to break.
Third, They get huge and grow unbelievably fast. These fish are one of the real champion growers out there; the farm I was working with saw growth of up to an inch a week.
As far as I can tell, they are legal to possess. They're just poorly suited to captivity unless the above factors are considered. The ones that I was going to introduce into the hobby would have been pellet trained. The Asian fish go for sale because the marketers buy them from Osage Catfisheries in the states or a few of their dependent farms in China. Once they're bought from OC, its up to the marketer who bought them to do whatever they want. No one stateside seems to feel that they're worth selling as pets here.
Another thing I'd like you to consider is this: how often do you actually see a paddlefish in those Asian keepers' tanks that are more than a few inches? If THEY can't keep them alive well, that's fairly telling IMO.