Definitely so. Unless you are on the equator, the sun is up for less than 12 hours a day from the autumnal equinox to the spring equinox. The sun will set before 18:00 local solar time. So apart from funkiness with time zones and summer time (which extends a couple of weeks past the autumnal equinox in Aus), yes, roughly half the year.
You're not crazy in the broad sweep of your idea, but actually because the sun isn't a point of light you're also not strictly correct, for example in Singapore the day is always more than 12 hours long.
Singapore is close to the equator (1°17′N) — the days are roughly 12 hours long all year. No, it's not exact. They vary from +3 to +12 minutes. It's close enough (<2% error).
It's not entirely due to the apparent size of the sun — refraction due to the atmosphere has a slightly greater effect.
(Singapore is also in the 'wrong' timezone. The sun sets around 7 pm every day, giving it effectively permanent daylight saving time.)
But regardless, Australia is not near the equator. The timezones are mostly ok. In most of the country (for most of the population anyway) the sun sets before 18:00 for roughly half the year. No amount of solar is going to power the evening peak demand without storage.