The philosophy is to announce their message on the blockchain, which is considered a terrible advertisement IMO, but who cares. They spam money, I wish they start spamming on my addresses too.

A burn address is considered to be a "proof" that no one generated. For example this address: 1111111111111111111114oLvT2. You know that it's a burn address because if someone had the private key, he could have the private key of any address (due to possibilities). You have 1 in 58⁶ possibilities to create an address starting with 1BURNED which isn't impossible. It's actually very easy even for an average CPU.

What do you mean by saying they don't have a private key? That there are no generally or that no one knows them? Every address can be created by around 2⁹⁶ private keys. There is no proof that the above addresses have a private key, it's just highly likely to have one. And not just one, but around 2⁹⁶ that none knows. Mind-blowing.