Hmmm...the title's a bit shortsighted (not wrong), and I think those two guys are more a scapegoat than a perpetrator. But here's my take on it:
The work DOGE did was the result of claims that were false to begin with. Not long after Elon Musk made his announcement that he'd cut billions in DEI programs when fact checkers took some serious doubts to that claim. Musk had to come up with something, so he brought in these guys, most likely telling and pressuring them to look into "anything DEI". I don't care about that manosphere myself, but I get why young rightwingers would believe that DEI is enormously wasteful.
If you want to blame these guys, blame them for their complete lack of critical thinking. They were encouraged and (again: according to me) pressured to come up with as much DEI as possible, and they just wanted to do the job as well as they could with the means (or rather: their lack thereof) provided to them. You can't ask or expect them to follow procedure when both their boss (Musk) and his boss (Trump) perceive the procedure as part of the problem.
Also...so they relied on chatgpt. So what? AI technology of this level is still relatively new. It's scary how a technology that was released in november 2022 got adopted, used and completely relied upon on such a massive scale (not just in this case, though this is most likely the most impactful). I've used AI for simple tasks that had an empirical answer (eg: how to set bazzite to use azerty in steam's interface), and about half the time, I got an answer that was demonstratingly false. Yet I've seen people rely on this tech for enterprise-kind of decisions, or even use it as a stand-in for human relationships. It's alarming, yes, but these clowns aren't the only one to fall into the trap.
Which trap? Asking a computer to search for biasses and use the result as if it's not the question's that's wrong. Asking an AI whether the holocaust is DEI ("start with yes/no") is about as absurd as asking the color of embarrasment, the sound of a stone or whether T-rexes were predominantly righthanded. It'll investigate, analyse and spew out a response based on what it finds on the internet. Meaning: if I write a short children's bedtime story about a lefthanded T-rex and chatgpt trains itself on it, it'll reply in that question with the same sincerity as if I had instead done a 5 year long study into dinosaur artifacts.