How about Waterhousea Floribunda? (Weeping Lilly Pilly)Native tree, mild root system I believe, and has lovely flowers followed by berries.
Also Eleocarpus Reticulatus (Blueberry Ash)

Rainforest trees like Waterhousea floribunda or, for example, Syzygium francisii, could be good, but 1. may depend on the rainfall in your part of Qld, 2. if you are so worried by 'messy' trees you may dislike the very ornamental fruit.

What about eucalypts? I think one of the most beautiful Qld trees is Corymbia tessellaris (Eucalyptus tessellaris), Moreton Bay Ash or Carbeen, which adapts to most Qld regions. Its roots are unlikely to be invasive.

Native frangipani is beautiful and not invasive. Illawarra flame tree is a bit messy being deciduous but the leaves are large and easy to pick up. I don't think its invasive either. Another tree which casts beautiful shade and is not invasive is paulownia. They are deciduous but have very large leaves and are easy to pick up, also if you have horses or other livestock they love the leaves especially when they dry out a bit after falling.

Technically, any large (or medium) sized tree has the potential to have invasive roots - it is just that some are more invasive than others. Structural problems are usually not caused by root upheaval, but by the reduction in moisture in reactive clay soils which cause soil shrinkage and potential movement in the foundations. It is usually a good idea not to plant trees and shrubs no closer to the house than their mature height - so, a tree that grows 8 metres should be planted a minimum of 8 metres from the house. As far as shady trees are concerned, you may want to look at the Blush Macaranga - grows to 6 metres and produces one of the best shade canopies of large heart shaped leaves I have seen. It self-seeds prolifically, however, so you need to be careful. Some of the grafted gum trees may be okay - 5+ metres high with very showy flower displays in summer.

Ian