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Bottle brush trees/shrubs from the genus Callistemon, I would not consider as having highly invasive root systems. But in Australia you have Melalueca species, some with a bottle brush type flower, which may be larger growing and more invasive.

Keep in mind that tree roots seek moisture, so planting in close proximity to drain tiles or an aging and possibly cracked sewer line may present more likelihood of root encroachment.

https://www.gardeningknowhow.com/ornamental/trees/tgen/trees-with-invasive-roots.htm

I have not been able to identify your tree as any variety of Bottle Brush.
I'm also not able to find any Bittle Tree that you mention a garden center identified your tree as.

Your leaves have more of a fruit tree shape.

The image that show what you describe as white stuff is too blurry for a good id.
This could be Mealy Bug, Wooly Aphid or Scale.

I would treat the plant with Neem Oil, it may take multiple applications.

You can try your local County Extension Office for an id.

https://www.gardeningknowhow.com/extension-search

Wind and cold can lead to leaf scorch, which causes the ends of the leaves to dry, turn brown and die back.
Brown, dry and crumbly leaf edges indicate drought stress. Water weeping bottle brush once weekly during dry weather so the top 8 to 12 inches of soil remain moderately moist.
An iron deficiency, called iron chlorosis, causes bottle brush leaves to yellow and become stunted. The leaves eventually dry and die, followed by the death of entire branches and stems.
Few pests bother bottle brush, but armored scales can cause dry, dead leaves. These 1/8-inch-long insects have a hard armored shell and they congregate in clusters on stems and leaves. They feed on the sap in the leaves and stems, causing the leaves to yellow or deform. Entire branches eventually die and the remaining leaves look brown, dry and scorched. Pruning out badly infested twigs and branches with disinfected shears manages small populations.
https://www.gardeningknowhow.com/ornamental/shrubs/bottlebrush/growing-bottlebrush-plants.htm

It sounds like the top of the shrub died off and it's growing back from the roots.

You will need to prune the dead plant material and let the shoots grow.

hytophthora root rot is a disease that infects bottle brush plants that have been exposed to overly moist soil conditions. Under wet conditions, the fungus that causes root rot grows rapidly and infects the root systems of surrounding plants. Symptoms of infection include the yellowing of leaves, early defoliation, reddish-brown roots that are brittle to the touch, the die back of branches and the discoloration of branches or the trunk of the plant. In severe cases, root rot can kill the bottle brush plant.

The best method of control is to prevent the soil below the plant from becoming overly saturated in the spring and summer months. The application of fungicides available at local garden centers has also been effective in treating this disease. Apply fungicides per manufacturer instructions.
https://www.gardeningknowhow.com/ornamental/shrubs/bottlebrush/diseases-of-bottlebrush-plants.htm

Julie, It's hard to diagnose the die-back conditions from a photo, without knowing more about your soil and water management.
If I were able to inspect the plant and the growing conditions, the fist thing I would do is inspect for breakage that could cause one section/branch/leader to dieback.
Then I would dig down or use a soil moisture meter probe in the planting bed to assess the soil moisture content. If the rock has plastic under it then that could contribute to inability to water adequately. With high heat drought conditions that I've been hearing about in Texas, the water issue would be a big concern. If you think that could be it and the soil is dry, do a deep watering right away to save the plant.