Generally, if a private cinema opens up and stays open, it's because they've determined that they profit enough from their direct patronage that it's worthwhile. Since there's already enough incentive in place for this cinema to exist, I don't think it's necessary to get surrounding businesses to fund them. Although in an ideal world, they should get something for indirect improvements to surrounding businesses, but the issue is that there's no way to actually quantify how much benefit you get from it. The current system works well enough in this case.

On the other hand, if you open up a cinema that operates at a loss while propping up surrounding businesses by bringing people in, then your lemonade stand definitely owes them. If you have two competing lemonade stands in the area that only exist because of that cinema and one of them decides to fund the cinema to keep it running, then it increases operational expenses for that one lemonade stand, allowing the other to outcompete. It's a prisoner's dilemma problem. Everyone paying means everyone wins, while if one person doesn't pay, then they win regardless of what everyone else does. Pay your fair share.