True, but Byram, to me, doesn't even excel at the medium-range passes or any type of passing to kickstart the offense, which still is what you see far more in the NHL than guys skating the puck up. His outlet passing as a whole is very underwhelming to me for a guy that supposedly is a great all-around D and #1D in the making. He takes too long to make decisions, which both limits his options and makes the likelihood of him hitting that pass far lower than it could've been. Honestly I think that's probably a big reason why he loves rushing the puck because that takes that type of quick thinking out of the equation and he can just rely on his tools. Similar to Nurse who is great at rushing the puck up the ice and entering the zone, but is not a very good passer at all. Difference between those two of course is that Byram actually knows what to do in the offensive zone with the puck on his stick, Nurse does not, although Nurse skating is superior so he can more easily make those types of rushes.
I feel like you kind of have to be an elite skater like a Hughes or Nurse if you want to consistently be able to transport the puck up the ice. Also I've never been a fan of his backwards skating and feel like it gets him in trouble defensively at times. His mobility in all directions is very good as you say, but there's more aspects to skating than that which can be quite vital.