Under the DMCA(United States Federal Law: Digital Millennium Copyright Act) and its Safe Harbor provisions, yes, Youtube is protected from copyright claims, provided they comply when they receive a notice. But....the DMCA that gives Youtube (and its parent company Google/Alphabet) this shield is US law, but Youtube does large amounts of business in other countries, making it liable in those countries, which don't necessarily have the same protections. Case-in-point, the EU.
Under some other nation's law, Youtube may be liable for infringement if they wait to be notified. Alternatively, part of the agreement that was being hammered out between Youtube and the content companies might very well be some sort of indemnity agreement, that protects Youtube from the content companies so long as they uphold their end of the agreement.
Additionally, even if Youtube isn't liable for the copyright infringement itself, since they serve ads on such videos, they could conceivably be sued under the theory of "unjust enrichment". (Similar to the logic that the EU is currently trying to get Google to pay their "link tax" for Google News on).
There are also non-legal considerations to consider:
Benefit to Self: Youtube is now selling/renting copyrighted content such as movies and TV shows as well. By becoming a distributor, it's in their interest to not also allow free versions of what they are selling on their platform.
Cost-to-implement: As Ron Beyer points out, a call center/email center/mail room to receive the DMCA notices would be an enormous cost. What's more, it's a continuous, recurring and likely-growing cost. Youtube/Google is also not short on software engineers. It's much cheaper to pull some devs off of their current project for a couple days, implement a solution and call it a day, with occasional maintenance, especially as they are going to be paying those engineers' salaries anyway.
Business Relationships: Youtube/Google make most of their money from advertising (and off of user data, but that is often tied to advertising in the other direction). The content companies are often their customers. Angering your customers is generally not a good idea.
Ultimate Desires/Future View:Youtube doesn't care about the lost content. They don't care about the fight. If the copyrighted material is blocked, their users are far more likely to find something else than leave the site; if the material is blocked before uploading, most users won't even realize it was ever there. As such, Youtube doesn't care if copyrighted content is blocked, as that is not their main market.