The easiest place for a backdoor to be injected is at the chip fab, after a chip is designed and taped out, before it is written to silicon. Several of these have been detected via electron scanning microscopes, so it does happen. But IMHO it probably doesn't happen to most chips. In commercial processors it can happen at the design level.

People can't produce fast hardware via open-source projects. But they can burn a circuit into an FPGA and be pretty sure there are no backdoors, but they end up with something really slow.

My thinking was "the most likely commercially available fast processors to not be backdoored would be new ones in new areas of technology".

Just because I think Intel and AMD chips have backdoors (sandsifter found hidden RISC instructions that bypass security, RDRAND acted strangely, Intel ME is pretty well known, etc) doesn't mean that the intelligence community is successfully backdooring even the early research projects. I suspect they aren't, but of course I don't know. Hitting the big commercial projects is a big win for them, but hitting every little research project is a huge cost with very little benefit.