Thank you for that illustration. That's what I was trying to get at, but this picture explains it so well. And even if that design did somehow work, why not make it wrap around 360 degrees?
Also, there's got to be a reason that it's not a thing (aside from Lucas oversight). Most lightsaber duels can be over in 5 seconds. Two blades strike; the person who first slides his blade along the opposing blade and to the hand wins. What would a cross guard accomplish that isn't already accomplished with conventional lightsabers?
And while super-friction sounds like a good enough explanation, there is almost always the cool disengage motion where the blades are touching and then slide apart as the combatants whirl away from each other. One-way super-friction? Yeah, there's a reason why FORCE! is often a sufficient answer, just like MAGIC! That's the problem with taking traditional swordfighting and applying metal-melting properties to it.
I'm still going to chalk it up to "It looks cool but we have no idea how to make this work."
The new lightsaber fails at protecting sliding down to cut off fingers, because the beams don't come directly out of the other beam: