Hey Lindsay,
"Or will geodesic fix this too?"
Yes, Kind of.
In the example above I've calculated the area in both Planar and Geodesic.
When calculating the planar area of the data in Zone 51, the geometry is projected from the globe to a flat extended plane which is centered on the central meridian of zone 51, somewhere to the east.
This is why the data gets distorted the further you are away from the central meridian of the zone and why calculated planar area for Zone 51 is larger than Zone 50 (roughly where the source data is).
For the geodesic calculations, the geometry is still being projected to the zones, but now the extended zone wraps around the globe rather than being a flat plane. Since the GDA zones are uniform, the area and length calculations of features will be the same for geodesic measurements between zones.
This doesn't negate the need for zones however as Pro map views also operate in planar space unless specific tools or operations provide the options to create or calculate data differently. (e.g. create geodesic in editing, geodesic viewsheds, calc geometry in geoprocessing etc.).
The mind bending thing is, like projections, all these area calculations are mathematically correct. You just need to use the one that is fit for purpose and record it when reporting. For large area calculations within Australia you can also try the GDA 94 Geoscience Australia Lambert projection or Albers.
(also upgrade Pro, if you are not seeing the dropdown options)