I believe you're overthinking it.
A Tinker tinkers with technology and uses it as a weapon. Their primary fighting skill is using technology. That doesn't specify how much of that technology is theirs, as long as they know how to build and repair their weapons.
A hunter putting a scope on his rifle isn't a tinker, but a draenei building a mech using naaru crystals or a blood elf building a manned arcane golem are.
The way I see it, there are three "levels" for this:
1) Like Demon Hunters, only one race per faction: Gnomes and goblins can be Tinkers.
2) Like druids, two races per faction: Gnomes, draenei, goblins and blood elves can be Tinkers.
3) Like other classes, most races except for a few: Gnomes, draenei, humans, dwarves, goblins, blood elves, orcs and undead can be Tinkers.
Considering they'd probably have custom visuals depending on their race, I'd say 3 is unlikely.
On the other hand, they could make all Alliance Tinkers use gnome technology and all Horde Tinkers use goblin technology, in which case they could go the DK/warrior route. But I think that has a lot less personality.
I'm fine with either 1 or 2.