When trained, they get sent to LVL 26 catas along with regular troops.
They get the same Off/Def bonuses as regular Troops. (Bar unit XP cos it's not in place for them yet.)
They get Hero Off/Def bonuses - SH Troops don't.
They can't be sent to SH Bgs.
They can't be sent to SH Hamlet's.
They show up in Cata's along with regular non-SH Troops.
A Brigadier Champ doesn't reduce their build times.
They don't show up in the number of SH troops you have in the Embassy | Stats tab.
The only reason the building is an SH one is because a lot of players wouldn't have room to place a new building anywhere else.
Using your reasoning; are Champs also SH units?
So to call them SH troops to justify why only SH boosts can be used? Not a hill I'd want to die on.
To sum up; you'd expect all normal boost options (Dragon, Market, Sapphires, Regular Daily etc) that apply to non-SH troops would apply to these (also) non-SH troops.
But I guess the answer just comes down to this.