Hi @chris_cummingsDT,
I had a look into your design block document and I agree with @jorge_garcia's assessment. Most of the air-wires are caused by the wires not being connected point to point especially the thick ones on bottom signal layer.
Regarding your comment: "I guess I am wondering why polygons in a PCB design can be assigned to a Net but polygons in a component cannot be." - the answer is simple the nets (signals) are only present in the design and they don't have any meaning in the Library, the signals are containers of connected objects (wires, pads, vias, polygons, etc.). The polygons (polygonal regions) in the Library are polygon shapes that cannot be poured and they don't have any connectivity properties. The polygon pours are only available in board design and can be poured around any other objects.
Regarding the copper pour not blending / overlapping this is caused by the fact that you have allowed polygon thermal relief connection for all pads / smds in the same signal as the polygon. When drawing the polygon please turn off the thermals, like below:

The Thermals option needs to be set as shown above. Now because you have already placed the polygon pours if you click on the dotted polygon outline you can change the Thermals option using the Inspector like it is shown below:

Uncheck the Thermals check-box and this will give you direct connection of the polygon pour to all pads / smds of the same signal. After changing the thick botom layer wires to polygon pours the number of air-wire errors has dropped from 13 to 6. The +5V thru-hole via in the middle is also not connected point to point and after routing an extra small wire from the via center the air-wire error for this via is removed. All remaining air-wires have the same cause.
I hope this helps.
Please let me know if there is anything else I can help you with.
Kind Regards,
Constantin Popescu
Principal Software Engineer