Because of mismanagement.
Like any other budget. I live on the land. For water, I'm "off-grid". I need to manage inflows, outflows and storage.
The power network is similar. We manage what goes in, what goes out and storage. If we run out, then we've mismanaged. Time to find better managers.
Baseload: taking a shortcoming of centralised generation and making it a perceived necessity. A tribute to the art of spin!
Something similar that seems to have dropped out of fashion recently is inertia. It's basically a characteristic of the large spinning mass of old-fashioned generators. The upshot is that it stabilises the grid. For a while, politicians and their spin-meisters were touting inertia as a necessity like baseload. For inertia, these days read Frequency Control Ancillary Services or FCAS. Pumped hydro does FCAS better than gas turbines. Batteries do it best of all.
LCOE relates only to generation. Storage doesn't generate power. What it does is store what would otherwise go to waste. What comes out may be less than what goes in, but all that comes out is savings. Storage is a grid service. It isn't related to any single source of generation. If you consider the grid overall, then its LCOE without storage will be higher than with it.
A traditional centralised generation grid needs backup; at least enough to cover failure of the largest generator. Traditionally, that's taken the form of "spinning reserves" - turbines kept spinning, but not generating, just in case. That's expensive so, more recently, various types of gas turbine have become more common. That's what failed in South Australia. The turbines weren't spinning, they took too long to fire up and the grid went down. That's why SA is going for the battery instead of more gas turbines. The battery is comparatively small, but the perturbations that it needs to handle are in the microsecond range - far beyond the reaction times of any turbine that isn't already spinning. By the way, if more than one large generator fails in a traditional grid, the grid generally goes down.
Nuclear generators are large. Assuming we go for storage as the reserve, rather than more expensive traditional types, going nuclear will probably demand more storage than will distributed renewables, each of which is comparatively tiny.
As to storage, in addition to batteries and the 22,000 potential off-river fresh water sites found by the ANU, there's also the option of salt water pumped hydro. Most of Australia's population lives on or near the coast. Along the coast are lots of cliffs and hills. In the ocean is - well, you get the idea. The concept proved rather successful at a pilot project in Okinawa. Originally intended to run for only five years, it was decommissioned after 17.