Would solve a lot of problems.
Class specific gear would require a token system to work. I think the problem this phase and possibly going forwards is that Plate/Mail (war/pal) only has competition with the same classes. Leather (Rogue, Hunter, Druid, Shaman) has to deal with Plate/Mail plus the other classes that potentially want their gear. And finally cloth (Priest, Lock, Mage) has to with Druids and Shaman competing.
I really dont anticipate this to change going forwards as dps plate will still look at mail and leather, mail will still look at both leather and cloth, and druid casters will still dip into cloth. Ultimately it comes down to RNG with the drops.
“problems” seems like a pretty strong term here to use in a game people are playing for fun.
To be fair though, if your biggest problems have to do with what loot dropped in BFD, you might just have everything else so figured out that you’re doing really great otherwise.
Out of curiosity, what role do you personally play in the formation of the BFD groups that you join, and what insight do you have that you could share with how they are put together?
Retail. 
Just don’t invite anyone who is going to roll on gear you want.
I mostly run with guild. I have done 3 pug groups. 2 went well, 1 was a disaster. I have run 2 or 3 GDKP’s. The GDKP’s put the most emphasis on comps. My guild run always has 3 to 5 clothies and maybe a resto druid. I can’t seem to get the gloves or the ring but I do have 347 GS or something. (but we do know GS is broken but I suppose it does have some value). It would be impossible to not include ppl that could need the gloves or the ring. If there was plus healing only and plus only damage, it would make the need more distinct. And I think every boss should drop something for each group, It should always be that people can only equip one type of armor. Example, paladins and druids should NEVER be able to equip cloth. I was in one pug raid where a ret pally wanted to need on all the cloth items but they did stop him with MS>OS.
Example: In classic that one boss would drop Benediction quest. ONLY priests could pick it up. (it was like 50% drop and I forgot what the other one dropped). My guild had a pecking order of which priest would get it, based on your runs. Therefore, someone brand new to guild raid would not go ahead of someone that made every run (I was 2nd in that line). And the priest leader had a meeting and we all agreed on the pecking order. This way insured that ALL priests, either shadow or healing, would get Benediction (Anathema for shadow) if they ran with the guild. At the end, we were bringing in alts to get it.
It’s called : personal loot .