A governing body does not define a notion. It defines specifics, that specific in this case is, what is a sport. This needs to be defined so that the governing body only gives support to the people that meet the criteria. In our case a very clear definition is available and we do not meet it. Simple as that.
They are not "just some governing body" they are the people that are basically responsible for sport in the UK. We are detached from them because in comparison to some of the other sports that they deal we will look not even close to becoming a sport. There are no governing bodies anywhere in the world(as far as I know), there are lots of different versions of the game (not always bad for example skiing and swimming have variations, however they are specifically defined in rulebooks, which are controlled by governing bodies) and the VAST majority of paintball players do what most people would describe as a military simulation.
Compare that to other sports, with national and international governing bodies.
These bodies have definitions for all the equipment, the techniques, safety standards, for all the different defined types of a specific sport they allow. They are the ones that allow the teaching of official coaching/reffing qualifications. If new technology came out they are the ones that will say if it is allowed or not in competition. They are the ones that change the rules if they think it benefits the game. In paintball this is all determined by individual event organisers, who can do almost anything they want.
Just in case you think it is because I am just not passionate about playing. I have played numerous tournys nationally and internationally over many years, I have spent many thousands on paintballs, transport to events/training, entry fees and gear. Then due to a severe eye accident at a paintball event when unsuitable safety precautions had been taken by the event organiser, I needed surgery, months of trips to doctors and even now it is not perfect. After all that I still play as much as I can.
If you want to try to keep to some subjective opinion on what a sport is fine, BUT every recognised sport in the world works with governing bodies, objective rules, definitions and minimal if any subjective opinions, e.g. degree of severity of a tackle.