In the city I grew up in in Canada, the nearby community centre was a relatively nice building on a big piece of land surrounded by a park and playgrounds. Not every one, but many had large halls and gymnasiums, some were collocated with schools, some with art galleries, etc. In the winter many distributed sand and salt, and they just generally acted as hub for anything the community required and anything the government needed to provide to the community. They also served a purpose as a shelter and support site in the event of a natural disaster.
As a child of a relatively poor family, I spent a fair bit of time there because the local government sent us a catalogue a couple times a year detailing all the programs they were running which due to our income level were more or less free (being a whopping $30 or something otherwise).
Based off this and your other comments, it seems a more visceral reaction to the phrasing “community centre” than that you have any real argument against them.