Last time our attorney attempted to explain this to me, it was along the lines of "the law does not allow us to question the DMCA" and that we have to take action, and it's the responsibility of the accused to file a counter-notice (all they need to do is send a properly formatted counter-notice to the same dmca@ email). If we play a role in determining validity, then we become liable and can be sued for damages just as much as the person who posted if we turn out to be wrong. We can't put the entire network at risk to try and do what we and our users would argue is right. That would be fairly irresponsible on our part.
That's not to say we don't still question really weird takedown requests. I've sent at least a couple back to legal seriously questioning what the link to the infringed content had anything to do with the content on our site, and some clarification with the sender revealed it was an error and invalid. But if they had insisted that it was correct and action should be taken, we likely would have still had to take that action.
The only part of this we can improve is the transparency, and it's something we're still working on. We've recently been cooperating with legal in our move to the new support system to make sure DMCA requests remain visible to us at all times, because we're eventually going to start pulling those out of the support system and into an internal archive for safekeeping.
One key feature we want to add is a new lock reason for the question which better highlights that it was deleted due to DMCA and who sent the request, rather than having an unnoticeable comment that's sometimes hidden. This also would prevent the question being subsequently undeleted if the staff member who deleted it ever left the company, or if moderator diamonds ever stop preventing undeletion by the community. It might also provide a way for us to directly link the post to a specific DMCA notice so we don't have to go searching for it later, if need be.
Once that lock reason is in place, we'd retroactively apply it for all the posts we've deleted by simply searching for all the comments/flags we've left describing the DMCA deletion. Having all these locks would allow us to just look for posts with the lock and easily run numbers to generate transparency reports for how many posts/accounts were affected without any additional effort on our part.