Chile, I thought your barbacoa recipe was with chucks but I just looked and you did it with butts. I wonder why I did it with chucks. Anyway, the only thing I didn't like is my corn tortillas kinda fell apart b/c of the liquid of the meat. I read the other day if you put cheese first that it will help keep the shell from tearing.
I've done 'Barbacoa' with various cuts of lamb, goat, beef and pork.
I've used beef Cheeks (favorite), Chuck's, overcooked Brisket, pork Butt/Shoulder, and goat.
In Mexico, depending on region and availability, many types of meat are used and they use all the cuts and some organs of the animal being used.
Barbacoa is a method, i.e. pit cooked, very slow, very moist and almost as tender as butter on the palate.
We've adapted our bastardized version to include smoking then braising for a smoky succulence.
As to your tortillas falling apart.
Drain the meat.
Are you steaming or toasting them first? Either is good.
If you look at my tacos I always double up, and only use corn if possible.
In Mexico there are literally over a dozen types and thicknesses/sizes of taco tortillas that are meant for different tacos.
Cheese?
Neither soft or crumbling cheeses are a traditional topping or condiment for Barbacoa tacos.
Various salsas are the mainstays, with perhaps a little onion/cilantro.
But hey, I'm guilty.
I like a little queso Anejo crumbled on mine.
Do what you like, if it tastes good, eat it!