That might be how the system worked if YouTube actually followed the law to the letter - instead YouTube actually preempts most of the DMCA claims by aggressively using Content ID to pre-flag content as soon as you attempt to upload it to their system. This approach might be better for users as a lot of content owners just assume that Content ID will catch everything they care about and thus file less post-release claims letting content creators mostly pre-vet whether a video will cause a flag on their channel buuut - it doesn't actually stop anyone from filing DMCA claims anyways.

I think it's important to distinguish what the law says and what is actually going on in the world though - since the actual DMCA system is only very occasionally relevant to creators (and usually tied to soft claims - like reading out copyrighted material or static content - like images).