How much concrete is required to anchor down a gazebo?
First, I should disclose that I voted to close as opinion-based. The answer depends almost entirely on your tolerance for risk and your effort/cost budget. Anyway, here are my thoughts. 


A structure of that lightweight design doesn't call for concrete. It'll be torn apart by high wind before it'll pull out good soil anchors. 
The anchors that came with the kit don't sound adequate. I'd purchase some larger screw-type soil anchors that penetrate at least 10 inches and have good bite. 
If your organic soil is very deep, consider digging through it and installing the anchors into the subsoil. Use chain or wire rope to extend the loops and connect the gazebo to that.