Why dont quivers have sockets?

Yeah 3+3 is significantly worse than 6. It would be interesting to share quality on supports, but wouldn't make sense in the same way the game uses your highest lvl active gem.

More detailed breakdown of 6 vs 3+3:

For pimping out 1 skill:

3+3 = 5

For 2 skills:

3+3 = 4

It's difficult to balance this perfectly; one side or the other will end up with at least a situational advantage. 4+4 seems like a good compromise though: For 1 skill 4+4 = 7, for 2 skills 4+4 = 5.

EDIT: Pictures! Just because I can.

Two skills, let's call these skills (X) and (Y). [d], [e], [f] etc are supports. = sign is a link.

Two 3 socket items, 1 skill:

(X)=[d]=[e]
(X)=[f]=[g]

This is equivalent to a 5 slot:

(X)=[d]=[e]=[f]=[g]

Two 3 socket items, 2 skills:

(X)=(Y)=[d]
(X)=(Y)=[e]

This is equivalent to a 4 slot:

(X)=(Y)=[d]=[e]

Oh right, I forgot to mention that, it's pretty big.

I found your explanation to be very unclear though, you might want to improve it for others to understand.