How to I fasten this toilet flange down?
I would drill out and break out more concrete to develop a hole that would hold a new concrete plug/doughnut around the pipe, rather than the difficult pointy cone you have now.

So, drill from the flat surface you have, lots of holes, and chisel carefully between holes.

My personal experience with Tapcons and "hard concrete" is abysmal. In green (new, fresh) concrete they seem to work OK most of the time.