Hmm... this is an interesting question, as it's not YouTube that uploads videos that infringe copyright, but they are indeed responsible. The mechanism at the moment can be abused from both ends:
Users can knowingly upload copyright videos, meaning that either YouTube or the Copyright holder have to find it, report it and take it down. The uploader was the one that broke the law and YT or Copyright holder were the ones that took action.
A user uploads a legitimate video, but then a lazy agency (on behalf of a copyright holder or YT) flags a video when there was no copyright infringed.
Both of these things happen all the time - I've had lots of "take down" notices handed to this site, and 95% of them have been agencies that have taken action on lazy searching and submitting batches of pages, because in most cases that search would return copyright content (but in our case doesn't). I imagine this happens on YT too (i.e. someone searches for a film title + a few keywords, then submits a list of all those videos to be taken down). Action is taken very quickly, with minimal consideration or checking taking place - as there is so much content to police. This results in lots of legitimate content being taken away too.
Sadly, I think we've probably got a "least worst" solution in place at the moment - as it could easily be too heavy handed, or not enough being done to protect copyright holders. It just needs to be a bit more robust from both ends - i.e. sites should take reasonable and proportionate steps to prevent copyright material being uploaded, but claims can be countered quickly with a manual review taking place (rather than just ignoring counter claims as they're too time consuming to police).
I would vote "joint" responsibility on the poll . So if someone reported the video and YouTube took no action, then yes they are responsible. Larger sites also have access to technology to use AI and tech to assist in this, which does a lot of the hard work. However we shouldn't make it solely their responsibility, because it would mean that we have to pre-moderate everything online. That could end up with some rather worrying scenarios that no-one would like to see.
@floppybootstomp - I don't think they should have taken down that video, it's not like you are monetizing that video. @Ian - I think YouTube should invest some of their millions to have a human monitor uploads so that copyrighted material doesn't get published. It would severely slow down the process but it would be better for everyone.