The exclusivity of the domain name is a public good. The right to use the domain name is a private, transferable good.

Were the auction site operated by Romania, the value of courses.ro would be determined by auction, paid by a willing buyer, and received by the Romanian polity. Instead, the Romanian polity receives 300EUR as an entrance fee to a race-track where luck is significant in determining the winner. There is nothing in the linked article suggesting that the number of entrants is limited.

It's an excludible good in the economic sense, but the value wasn't created by the private company, but by the public WWW itself. This is classic rent seeking.

Sure, just like the public's promise to keep off a chunk of land is a private good.

That's how our current system works, but it probably works that way primarily because the groups in charge wanted to raise money, not because it's a fundamental philosophical truth that it should work that way.

Can you clarify what you mean by that? Would you say that ICANN "owns" all domain names, then sells them in bundles, making them private goods?

A domain isn't a private good until it's actually owned by a private party.

"A domain name is a private good, not a public good.... until it is sold (or rented) to person x, with the understanding that he can renew it forever if he follows a few rules.

Country ones are owned by the country - TLD's are a bit more unclear.