Yes, it is possible for the original person to provide documents showing their ownership, and for the company to put the content back up.
It has the potential to be a decent piece of legislation. It removes friction for sites to allow UGC, it allows copyright holders to issue takedowns, and it allows for companies to make decisions about contested takedown requests if they care enough to do so. However, users who upload content abuse it, sites who know they have a lot of infringing content abuse it, and people without claims abuse it by issuing false takedown notices. The law would be better if takedown notices, their resolution, and most things about the process were transparent, and there were penalties for users who upload content they don't own, penalties for companies who infringe far more than they don't, and penalties for people who issue takedown notices without a legitimate claim. I believe an administrative organization responsible for tracking and reporting these things is worth the expense to let companies keep innovating while protecting the rights of content owners, and making it easier to penalize abusers of the law.
It has the potential to be a decent piece of legislation. It removes friction for sites to allow UGC, it allows copyright holders to issue takedowns, and it allows for companies to make decisions about contested takedown requests if they care enough to do so. However, users who upload content abuse it, sites who know they have a lot of infringing content abuse it, and people without claims abuse it by issuing false takedown notices. The law would be better if takedown notices, their resolution, and most things about the process were transparent, and there were penalties for users who upload content they don't own, penalties for companies who infringe far more than they don't, and penalties for people who issue takedown notices without a legitimate claim. I believe an administrative organization responsible for tracking and reporting these things is worth the expense to let companies keep innovating while protecting the rights of content owners, and making it easier to penalize abusers of the law.