There needs to be a balance
If there is going to be pressure (fines, or political threats) on platforms to remove illegal content, there have to be equal threats to them if they remove legal content. Otherwise, as the commission appear to acknowledge, the obvious impact (remove anything when there is any doubt at all) will happen.
Platforms should only be removing illegal content. So, if content is removed there should be a right to challenge the removal in a court. If the court determines it was illegal then you get hit with a significant fine paid to the platform for posting it. On the other hand, if it was not, the platform gets hit with a significant fine paid to the submitter for removing it when it was not illegal.
Make the two fines high enough and the system will not be overloaded. Although only rich people will be able to afford to take the risk involved in the challenge, it will at least give the platforms a business reason to invest in properly functioning determination processes, which should feed down to all of us.
Some may argue that as private companies, platforms must be able to remove anything they want to. I say that when acting as an arm of the government (under the sorts of threats made in this paper) they lose that right and have to accept any and all legal postings.