Happy to stand corrected, but this is how I read this paragraph:
the carriage of persons or the carriage of goods without a charge for the carriage being made other than the carriage, for the purposes of trade, of goods being the property of the pilot, the owner or the hirer of the aircraft;
I interpret this as the
other than applying to the
charge not the goods i.e no charge being made other than the hire of the aircraft (i.e no cargo rates or pilot wages etc). So the carriage of goods for trade is private, as long as the goods are owned by the pilot/owner/hirer and the only charge is for the aircraft hire.
To read it as "carriage other than for trade" the comma would need to shift ie:
"the carriage of persons or the carriage of goods without a charge for the carriage being made, other than the carriage for the purposes of trade, of goods being the property of the pilot, the owner or the hirer of the aircraft;"