I can understand if it were sent on the wrong network, or with the wrong memo, but the request here was to transfer ETH on BASE, they provided a BASE deposit address, and I sent the funds on BASE through Coinbase wallet which is not exactly an obscure wallet. In fact I think BASE network is run by Coinbase.

So obviously they should be able to handle this or, at a bare minimum, refund the coins since they control the address. Or give me the private key to the address and I'll do it myself.