As far as I can tell, usually what you say is right, each airline takes a certain number of seats to sell and from the availability you post, it looks like Lan's Business class allocation has sold out while Qantas still has availability. Looking at the seat maps in EF can sometimes give an idea of which seats are allocated to which airline. Of the 42 Business Class seats in the cabin, the LA seat map shows 36 while the QF seat map shows just 6. Since the availability for QF321 says J6, this implies that Lan have sold (at least, with possible over-selling) all 36 of its seats while QF have sold exactly zero. This could be price-related as you point out, but it seems a bit of a stretch that not one loyal QF customer would have bought a ticket on the codeshare while ALL the LA seats have sold out.

If you do a search +/- 3 days in EF, there is heaps of availability on every other day but the 22nd. I'm completely guessing here, but it looks like unfortunately a group has booked out the entire Lan allocation in the Business cabin on the day you want to fly (sports team perhaps?)