Sure it does. The creature exists once it enters the battlefield. It goes on the field, then its ETB goes on the stack. If it’s countered, the ETB does not go on the stack. Therefore, its ETB is reliant on it existing on the field. Once that resolves, its secondary ability triggers if you put at least one of those counters on opponents’ creatures.
The Game Boy line of handheld consoles revolutionized the gaming industry and were fascinating devices, even if they had some absolutely bizarre models and accessories at times. Did you know there existed (but was never used) a Game Boy Color accessory used to anesthetize children before surgery? True story.
But if the paper nation flanks us we’ll be at a stalemate. We have a conundrum.