Their responsibility begins with removing the hosted data in question, where a valid DMCA notice is issued (and there is no incentive for them to make a judgement about whether the notice is valid or not.) If the authors of youtube-dl want to file a counter claim, that is their right, (which would put GitHub within their rights to restore the content too.)
https://www.dmlp.org/legal-guide/responding-dmca-takedown-no...
All of these declarations are made under penalty of perjury, both claims and counter-claims:
> The DMCA requires that you swear to the facts in your copyright complaint under penalty of perjury. It is a federal crime to intentionally lie in a sworn declaration. (See U.S. Code, Title 18, Section 1621.) Submitting false information could also result in civil liability—meaning you could incur a financial penalty.
The civil liability here is a liability to the party who was damaged, (the author or copyright holder), so even if GitHub wanted to assert by themselves that the DMCA takedown claim was invalid, they would not have standing to sue anyone about it. So pretending even if you did believe youtube-dl authors are in the right and that the courts would be inclined to rule in their favor, and you're Microsoft, you have to honor the DMCA request and take down the content in order for there to be justice, since there can be no party with standing unless there are actual damages. (IANAL, you probably already figured that out by now, and I have no idea what the legal definition of "actual damages" is, but I do know what standing is.)
Personally I agree that this does not represent a valid DMCA claim, but for GitHub to assert that and ignore the claim based on the way these laws are written, and those safe-harbor laws as well, I think any lawyer would say this is not possible.