The Tonys don't have to cover everything. An audience member cannot judge stage management, dance captain, swing, music direction, casting, etc. by merely sitting in the audience, which is the purpose of the Tonys. You also cannot have any categories that require voters to go to a show more than once (such as Replacements); turnout is tough enough as is.
If anything, leave it up to the unions and guilds to devise their own awards like with the film world (SAG, PGA, WGA, DGA, ACE, Artios, etc). But the unions won't want to exclude tours, regional productions, off-b, etc. since the majority of their members don't actually work on Bway.
To pick apart two oft-cited examples:
On a musical, you might have different people in each of these positions: Music Supervisor, Music Director, Vocal Arranger, Dance Arranger, Conductor, Orchestrator. That's not even factoring a Composer, if they are active and alive. Each of those roles is integral and if they're done well, it doesn't matter to the audience who did what. If anything, evolve the Orchestration award into one that covers both Arrangers and Orchestrators. But as with the Orchestration award, it would probably just go to the show that wins Best Score or the show with the biggest orchestra.
With a Casting award, there's the issue that many roles get cast without the involvement of a Casting Director (especially stars). The casting director is not making final choices on who gets hired. And then it's the Director's job to bring out the performance from the actor and make sure the ensemble is properly calibrated.