Communism is just late stage socialism. Besides even if they were two separate things, by design in socialism nearly everything is centrally planned. Corruption just spreads much faster in systems where you centralize power from the start ie you want the groups running the media, producing goods, and managing the laws to be separate entities
What you’ve just described are socialized safety net programs inside mixed capitalist economies. That’s not socialism. There is no private ownership in socialism. Consequently, there are no rich in that system. The “community” owns and produces everything.
