If this sportsbook is on a site with several games on Stake I guess they will get good traffic considering Stake has many active players there.
I'm okay if they decide to limit people from trying sites in beta because other popular gambling sites like Luckybit only choose a few people before opening it to the public. Besides, we can still try the beta version by following the giveaway in the Stake forum.
Other games do help attract more traffic, but they can only retain more players only if their odds are great with low margin and if their interface is not laggy and annoying like 1xbit(Easily one of the worst in this department).
It's happened before though on previous new game like Crash, before it released to public, it's open up for beta. I think it's not because of the traffic, maybe just for test to some people who win the beta access and finding bugs, make it perfect before it released to the public.
This is the problem right here. Crash and sports betting are completely different. If they opened up the beta to everyone, we could get a better idea as to how the sportsbook would function under heavy load(Cannot be assessed with a closed beta IMO).