Can find anything completely declarative on this subject, but everything I’m seeing suggests you’re right. Football clubs only pay for officers within the stadium, and it’s seems UK case law prevents the police for charging stadiums for officers one public grounds in the surrounding area [1].

There is also an ongoing debate about how policing costs of football matches are funded, given all police forces have seen huge cuts in budgets over the past few years. But equally football matches are much less violent than they used to be, and some question if the police tactics used today are the right approach (police seem to expect the worst, despite evidence suggesting their own tactics might be enabling bad behaviour).

[1] https://www.squirepattonboggs.com/en/insights/publications/2...