The agreements called for Russia to be the sole heir to the USSR.
Thus Russia inherited 

* The debt
* The properties abroad (embassies &c)
* Nukes
* UNSC permanent membership

This was the only natural and acceptable solution, similar to the solution taken after the British empire's dissolution. Just like it would have been absurd to make the UK share the UNSC veto power with the British Commonwealth countries (erstwhile colonies), it would have been absurd to make Russia share the UNSC veto power with the Commonwealth of Independent States.

Who would want a different solution?

 * Russia? Do you know of any country who would voluntary dilute its power?
 * The former soviet "republics"? They wanted as little association with Russia as possible, in fact, CIS was so loose it might as well not exist.
 * The West? They were so overjoyed with the end of the Cold War that they bent over backwards to welcome Russia "back into the fold"