That's a historical curiosity due to how the relevant sports developed.
Football, rugby and cricket were all formalized in the UK, so the initial arrangements reflected a vision of Scotland, England etc being "countries". The first international football match, for example, was Scotland-England.
As the game expanded, other countries started their own international federations, which UK clubs eventually joined. As part of the joining, British representatives obtained clauses that safeguard their role as "custodians" of the game, which basically mean you cannot change the rules of football without UK representatives agreeing.
Part of those clauses imply and require that UK countries will remain separate; so you have a Scottish authority, an English authority and so on, with separate leagues and separate national teams.
The two elements (representation in rule committees and in tournaments) rely on each other; renouncing one would basically mean renouncing the other. Of course, you're never going to renounce you ability to make the rules, so here we are.
In the Olympics, for example, it's different, because that institution was actually born in France and reflected a different vision from the start; which is why Britain traditionally does not compete in Olympic football.