Rich lists are impossible to produce with Monero.
Is it still possible to determine the total number private keys have ever been used to control available outputs on the network?
Obviously the number cold storage wallets that were created offline and have never been used is unknown. Counting the number of available outputs and combining that with prior transaction data can we somehow determine how many how many private keys control (or previously controlled) those outputs?
No, or it'd be a break of the crypto. Poloniex can probably get a lower bound (probably much lower than the real number), since it's the main Monero exchange and has been for a long time. But from the blockchain, no. Every single output on the blockchain is on its own one time address, and can't be attached to a standard address without that standard address' private view key.
Adding onto user36303's comment, the factor of the lower-bound can be estimated if we assume the majority of private keys were generated by the Monero Vanity Address Generator. Assuming the most people have calculated out of the generator is a substring of length 6, there would likely be some factor of 606 private addresses generated, so estimating this was done around 100 times would give you 4.67e+12 addresses.
The caveat here is this calculation has nothing to do with outputs on the blockchain, since these addresses were never even synced and effectively "cold storage".
