I plan on driving into Sydney from Katoomba on a Monday morning during mid-January. Would it be ostentatious of me to believe that I can see the harbour front, Opera House, and Botanic Gardens all in one day while still leaving time for lunch, dinner, and checking into an AirBnB nearby?
You should be able to do all that in the morning. There are tours of the opera house you might not have time for if you want to wander around the gardens longer - but they are all right in the same place, so if you have an entire morning you will be able to get a good feel for all three.
Get quick answers from The Opera House to the Botanic Gardens Walk staff and past visitors.