What you say makes sense but there's some things that should be realized. For one thing, hands on training has its limitations too. For one thing you can't always go full speed or full force while training. For somebody to swing a katana full speed and for you to attempt to catch it with a sai you can't do in training, simply for safety reasons.
As a martial arts practitioner, you don't train weapons vs weapons going full speed, unless it's a very specifically rehearsed demonstration done by two highly qualified people. Even then, they're going to practice slowly.
Trying to train otherwise with a sword vs sai at a real speed is just plain stupid, since someone will get hurt.
It's not the same thing as, say, two fencers or kendo-ka going at each other, since their weapons have a lot more margin of safety built into them. Even then, they're not going to be stabbing at each other with the intent to maim or kill.
Again, I'll state this, that trying to go at it full speed with weapons in an unrehearsed manner is stupid, and doing so with garbage mechanics sets a new benchmark for stupidity.
Sometimes a good way to know what a weapon was used for is by researching history, which is often done through books provided that you've got a good source. For instance, the jitte, a weapon that's similar to the sai except it has only one hook as opposed to two was apparently used by Japanese police and was used to catch and immobilize sword blades.
Again, this is a rather short-sighted view for the exact same reason I mentioned above. There would be so much momentum from catching a swung sword blade, that it would drive the tines into your hand, quite possibly breaking your own bones. It's nothing more than simple physics.
The user would be far more productive trying to catch the arm or the clothes worn by the sword wielder.
My source for that claim is from the Martial Arts Encyclopedia by Larry Winderbaum although I wouldn't recommend trying to back that up by getting a jitte and having somebody swing at you full speed with a katana.
Winderbaum's assertions simply don't hold up to simple science.
Think of it this way... Why don't motorcycle police try to block the roads when there's a criminal trying to escape with a SUV? Quite simple... The physics aren't in the favor of the motorcycle policeman. Even if a TV show demonstrates some super-powered policeman harmlessly deflecting away the SUV, simple physics do take priority in real life.
I would encourage you to take some courses in elementary physics. When you see the factors involved when it comes to things such as kinetic energy, you may be surprised at how small variances in the variables can result in sizable consequences.
The final answer is this:
You don't use the sai to trap sword blades in the silly manner that was demonstrated by the video. Period.