Since the article was short on answers, here's one possible answer as to "why we don't eat swans":

Their numbers are too small to get into volume "manufacturing" (CAFO's).

15,000 swans. That's cute.

We harvest 10,000 kiloTonnes of cattle per year in the USA alone. That means, our CAFO'S produce about 2 million pounds of beef for every 1 swan we could harvest.

Our food system loves repeatable processes & volume efficiencies that wild game can't provide.

Note: I may be nearly vegetarian, & eat only local organic meat rarely. But, the facts of CAFO's dominate the market.