And, well, sometimes it just simply is night, and sometimes the wind doesn’t blow.
Do you really think this isn't already taken into account?
We don’t have the battery tech to run from storage alone
Nobody is making that argument, as far as I'm aware. There are plenty of ways of storing energy, e.g. pumped hydro, that would work in conjunction with battery storage.
But, honestly why wouldn’t we use nuclear?
The obvious one. It's wildly expensive when compared to renewables, and that's before the usual nuclear build issues of cost and schedule overruns.