Even without this new system, a person setting a toy rocket off near a deployed military force is really asking for trouble.

I disagree. It is the deployed military force that is asking for and finding trouble. It is the innocent who bear the brunt of the danger for this. This is the moral hazard[0] of war.

[0] https://en.m.wikipedia.org/wiki/Moral_hazard

Possibly, though I bet the return from the cardboard/cardstock that most of those are made from are very different from the radar return from a metal missile.

I played with Estes rockets too. How many people are likely to in war zones like these APS devices will be deployed in are likely to though? The rocket motors aren't super cheap and aren't really reusable or easy to make your own.