Correct. Furthermore, my understanding of the Lightning Network is that for channels with < 2 recipients (other than sender) the fee is actually bigger than if the main network was used.... didn't really read much into the technical side of the LN though.
I am not sure what you are trying to say here. A channel is maintained between only two people. A node can have multiple channels. When some node routes a payment, it receives coins in one channel and uses some other channel to send the same amount of coins (minus the fee). If there was only one node which could route a payment from you to the player, it would not be necessarily more expensive than an on-chain transaction. The LN was designed for micro-payments, so you would end up paying a couple of satoshis or even less.
The LN might work if the player requested multiple withdrawals without sending coins between each withdrawal to an address/recipient not on the LN.
In other words, the player can receive payments over the Lightning Network as long as they don't close their channel.
LN contracts require a pre-payment, though, don't they? And a contract is between max two parties.
Meaning an additional amount will always be tied up in open LN contracts.