Timeline of the Great Rings

[Findegil]

Basically, the number 7 is connected to the Dwarves in a way that no other number (except maybe 12 and its square 144 in regards to Elves) is connected to any other race.

I disagree to this: There are 3 Elves that awake first as there are 7 fathers of the Dwarves. The 3 Elves are the first leaders of the 3 Clans as the 7 fathers of the Dwraves are the first leadres of the 7 Houses.
What more similarity could you wish for?

Even if you argue that the elves soon split into more significant groups (Eldar/Avari), still the split between Nulukizdin and Khazad has some similarity.

So the argument could be used rather the other way around: Since Sauron does know the number of the Rings (and probably influenced it) and is aware of the symbolic connection of the Elves with 3 and the Dwarves with 7, we might assume that there was a similar connection between Men and 9.
Since Sauron could not force his will through with the 3 Rings, he dedicated to the ELves, it is insignificant that they did not found their way into the hands of representatives of the 3 Elven-Clans. And it is rather significant that 3 of the 9 were given into numenorean hands: The Numenoreans were at that time the best representatives of the 3 Edain-Houses.

Under these assumptions we could gues that beside the 3 Edain-Houses we would find in the early history of Men 6 farther Houses. And for one of these 6 farther Houses we could assume a connection to Khamul, the Black Easterling.

Respectfuly
Findegil

P.S.: This does not mean that the Dwarves of Moria could not held more than 1 Dwarven Ring. Even in the choosen group of Thorin's companions not all were members of the House of Durin, so they all were descended from Dwarves of Moria (see the footnote to The Line of the Dwarves of Erebor in Appendix A to LotR. And as already said, the Dwarves of Moria were beside the House of Durin a mixture of all the other houses. So after the sinking of Beleriand and the decline of the Mansions in the Ered Luin, it could well be that the best candidates for Sauron to represent the Firebeards and the Broadbeams in reciving one of the 7 Rings would be found in Moria.