This is everyone's expectation in the short term, yes. Long haul would probably only be LCC, like Jetstar or Air Asia, maybe Scoot. There's no way Emirates or SIA (mainline) are moving here anytime soon. It's simply not where the high-yield and connecting passengers are.
It will be a slow start, the question is whether it ends up like Avalon or Gold Coast? All bets are it being more like Gold Coast simply due to the proximity to large numbers of residents who would provide domestic demand. But it won't be the same frequency of course, SYD will retain the lion's share for the next 10-15 years at least. Note that Virgin and Rex haven't even yet committed to flying at all from WSI, although if Qantas starts putting some capital city flights there, the others will probably follow. But if it's just a few Jetstars a day like Avalon, they may not feel the need to compete. It's a commercial decision for these airlines and there are many ways it could end up, outside the control of airport management.
The two airports are under different management and I don't think slot management / control at SYD has any impact on what goes on at WSI. There's no unifying authority that would manage something like that to my understanding (at least for domestic... international slot management is another beast).