I've had mixed experiences claiming from BA. I believe they deny some claims knowing that some people will give up. I also question whether they fulfill their obligations of informing passengers of their rights.
So I'm all for something that encourages BA to do the right thing. If people choose not to claim - that's fine. I've opted not to do so on occasions. But I bet many of the people who don't do so today either don't know their rights, or believe it will be too much hassle to claim.
I don't really see that 'Airline X is better/worse at....' has any relevance on whether BA should improve their conduct in this area, or indeed whether someone should sign the petition.