Yes, that's what independent
means. But it was de facto
independence, not de jure
. The Detapa Council theoretically had the power to control the military, but didn't, either through lack of will to do so, or because they lacked the effective means.
At any rate, the fact that all three entities were independent doesn't preclude the existence of a republic, (especially since it is clear that a republic did exist) and in fact Gul Madred's dialogue does support the idea that the military dominance of Cardassia's government is a relatively recent development that stemmed almost directly from their need for resources and apparent lack of them at home. Of course, without any control over the military, it's no wonder Central Command took over in the end, especially since they satisfied Cardassian society's needs while they were in power....