Sent ETC to my ETH address - Coinbase

I know it happened before to others but my issue is with Coinbase. I gave my client my eth address and they accidentally sent ETC to that address.

I find out that in that situation for ETH and ETC it's easy to retrieve the amount but only if you have the private key. Is that right or did I understand it wrong?

Coinbase has this policy of keeping your keys for you and not giving them. They won't retrieve that amount for me either.

Their support was kind of rude also. I asked my question and received an automated text template. When I asked about the ways I found that I could retrieve that amount and if they could help me with that they responded with the same text in bold. The third time I responded they just cut me off saying that due to high support traffic my messages won't be received anymore.

Edit: Coinbase account not Coinbase wallet