Actually, this isn't the full story. Military activities in a nation's EEZ is a point of contention: countries like Brazil, India, Pakistan, China, Iran, Malaysia, and Indonesia object to this. It's not explicitly defined in UNCLOS whether military activities should be permitted, and it's the prevailing view of countries that make up almost half of the world's population that it shouldn't be. International law hasn't really been extensively challenged in this regard until very recently (the past decade or two), so the debate on military activities within EEZ is still that, a debate.
Edit: for reference, it's mostly the big ex-colonial maritime trading powers that are in support of this because it makes their trade easier, while those countries who exercise coastal rights and natural resource exploration rights are opposed.