Welcome to the best race with totally no controversies attached at all!
Judging by what you described to be looking for, here are some from the top of my head that may be suitable.
Orcs of the Red Blade
Bit of self-promotion here, but ORB is an orc-only guild that strives to uphold the values of the Horde as embodied by Thrall after the Second War. They do tolerate all of the Horde’s races so long as their interests align in maintaining the safety and well-being of the Horde, but their own mode of life is exclusively orc-focused.
Frozen Paw Clan
A nomadic clan based on the ideals of the Frostwolf and Whiteclaw Clan. They do take in all races including Forsaken and all manner of elves (they’re the nice ones, though), but they too strive to uphold the ideals of the Horde and definitely have a family feel to them.
I don’t think they have a (recent) guild thread up, but give Rogmasha, Overhaul, Velravyn or any other ‘Overseer’ or ‘Second’ a poke.
Blood Howl
Granted, they’re sort of a warband, but they do far more than just combat-oriented events and, again, regard themselves as a sort of family that pursues the best interests of the Horde. They accept all races, but do lean towards the ‘true Horde’ sort of feel (if by ‘true Horde’ you mean the Third War kind of Horde).
Red Venturers
Again, sort of a warband (or rather, mercenaries) who accept all races, but they have the Horde’s best interests at heart. They style themselves as a bit of a band of misfits, so you’ll definitely find more than just orcs, tauren and trolls in there, but they still have a fellowship sort of thing going for them, I believe.