I agree with just using your pit regularly, the heat will keep anything from growing there. If you are worried about setting a root on fire just do like you said and dig down a bit and put in a layer of rock as a barrier.

Those look like the lead anchor type. You drill a hole push this combo in then pound the nail down to wedge the lead in the hole. I have removed some with a butane plumbers torch heating then pulling with vise grip. But be careful concrete pops when heated to much wear safety gear if you use this method. https://www.homedepot.com/p/Red-Head-1-4-in-x-2-in-Hammer-Set-Nail-Drive-Concrete-Anchors-50-Pack-35305/100158509