Bump YoBit is still ignoring me
They essentially have complete control over the private keys and as you said, it's up to them to sweep the funds out of the ETC wallet.
I haven't read YoBit's terms and conditions but it is most likely that the exchange does not own responsibility over their users' fault in sending coins or tokens to the wrong blockchain.
While it could be recovered, it probably falls on the platform's sole discretion whether to attempt to recover it or not.
10 ETH is a significant amount but I guess it is not worthy to start a costly legal process considering, first and foremost, that the fault itself is coming from the user. So I guess OP should just continue communicating with the site but expect less.