The ultimate question (besides my curiosity of how someone who would gamble away 13 BTC could have amassed such an amount) is if you were to have won dozens of BTC after your self casino ban, would you have been OK with Stake.com refusing to pay out the winnings because you should have been banned? I suspect your answer would be no. Certainly in that situation you would have a much better argument that would lead to more people supporting you.
Your hypothetical scenario shouldn't have happened in the first place.
Proper gambling sites don't allow a self-excluded account to be reopened, period.
If a user opens a 2nd account with an attempt to circumvent - then and only then - the user is 100% at fault.
I had a similar case with a much smaller website and they did pay me after I started legal proceedings against them.
Stake must not repeat that mistake and if they want to keep their reputation high they need to pay the user or reach a settlement. I am not sure why the thread mentions the user is a scammer here - the way Bitcointalk works is absurd.