你的判断错误源自于你疏忽了以下二点：
1 现在分词作状语，分词的逻辑主语可以是前面分句的内容，而不是句子的主语。这种分词多为结果状语。例如：
Tom lost his bike, making his father angry. 使父亲生气的不是汤姆这个人，而是汤姆丢自行车这件事。所以现在分词的逻辑主语不是Tom，而是逗号前这个分句的内容。如果改为并列句则为：Tom lost his bike and this made his father angry. 这个分词短语还可以改为非限制性定语从句，Tom lost his bike, which made his father angry. which的先行词为主句内容。
2 allow的逻辑主语并非必须是人，也可以是现象、体制、局面等。词典对allow的这种用法的释义及例句如下：
to make it possible for something to happen
Occasional gaps allow passage through the mountains.
a password that allows access to the system
所以你的句子中，allowing现在分词作结果状语，其逻辑主语就是gaming should be enjoyed in moderation, balanced with outdoor activity这件事，也可以改为which allows...这样的非限制性定语从句。
