The actual rule is that if YouTube takes something down in good faith belief that it is infringing, they are protected against any liability for any claim over

The actual rule is that if YouTube takes something down in good faith belief that it is infringing, they are protected against any liability for any claim over taking it down, but only if the follow the restoration procedures as specified. If they fail to do that they simply lose the protection of liability if you (or anybody) sues them over taking the work down. So you cannot successfully sue them over say lost revenue for a fraudulent takedown notice if they follow the correct counternotice procedure. If they don't, then they might be liable, but most likely something in the terms of service or similar would prevent your suit from being a winning case anyway. However, in other scenarios like where the provider was contractually obligated to keep the content available, if they fail to follow the counternotification procedures properly, you could potentially sue the provider and win.