The fastest way to make a team hate culture is to confuse play with recovery.
Play is not mandatory fun, and it is not rest. Play restores sociability through non-instrumental shared attention. It only works when it is not scored against work output, and when status can loosen for a while. The CEO can lose. The intern can win. That is play.
Rest does different work. It restores capacity by lowering demand: sleep, silence, daylight, a walk, a meal with no agenda. A tired team can sit through trivia night and stay metabolically clenched the entire time.
Get this wrong and you create theater: amusement when people needed decompression, or isolation when they needed contact. If you want renewal, provide both. Otherwise you are just extending the workday and calling it culture.