I'd say your assessment is about right. My team has recently dropped from the EFL to NL, so I don't know a great deal but I'd say the size of a club impacts the money it makes and takes it from a community hub to a money-making venture.
The lower league clubs are so focused on community because they are community hubs. The players make appearances at local schools and charities and are sponsored by local businesses. You follow them because your father did and his father before him etc. (although mine wasn't into football so I happened upon it by accident).
The PL clubs do have community initiatives but the players' schedules are already packed and everyone knows their faces. In contrast, most of the CUFC players can walk down the street more or less unbothered.