Will people seeing ex-DOGE employees on your staff (such as through LinkedIn) impact your company? For future recruiting? For business to business in your company's domain? For clients? I personally wouldn't work with people that operate in the manner DOGE has (push the limits and move as fast as possible so you can't get stopped, assume part of what you are doing is unacceptable/illegal and will be pulled back instead of making sure what you do is acceptable/legal) and I would be hesitant to trust a company that does unless I was looking to break things in an established space or something. Seperate from that the majority of devs I know wouldn't work with them for political reasons adding up to a whole lot of unfavorable to overlook if you have other candidates.
Do you recruit through current employees? Will tilting your recruitment pool to be more ex-DOGE centric impact your company culture and/or again recruiting other talent? I'm not calling them members of nationalist parties, but there's the whole 'let one into your bar and soon you have a bar full of them' analogy for a reason. Are you OK if you company is labeled an exDOGE shop or other political pejorative shop in the industry you are in?
There's enough other talent out their I'd just pass and less someone else juggle the factors.
I really hope they delay that rewrite and actually test it. They might, but if they don't…
As a hiring manager, know your team and your company. If you have to ask the question, you probably don't want them, unless you are willing to churn your team over to his ideological mates. If your organization was "dark, gothic Maga" then you wouldn't be asking.
However, I would be willing to bet there are plenty of potential employers that wouldn't, or would even consider it a positive. Unless things turn much uglier, I suspect having that on your resume would simply affect which set of companies would be willing to hire you, not whether or not you're hireable.
In that world, it will be an advantage.
But in just world, there would be actual investigation. DOGE is in fact breaking laws.