> it's just a standard case between two private parties resolved through our legal system

This is a gross distortion. Standard contractual rules bind the parties that signed the contract and the remedies are proportional to the damages and bounded. Copyright is tort law, the state binds the world to respect the rights of creators and the damages on infringement are punitive and can far exceed the actual commercial damages - to the point of bankrupting the infringer.

The key to torts is that the state is not neutral, there is a social good here it's protecting. Crucially, copyright, like some other torts - securities, antitrust, environmental, battery - also has a criminal enforcement regime, where, for particularly serious offenses, the state actually invests public resources to put the criminal infringer behind bars with little to no involvement from the original rights holders.

In the particular case of US, there is an entire state apparatus dedicated to enforcing US copyrights, a foreign affairs policy to shutdown "Notorious markets for counterfeiting and piracy" in other countries, international enforcement of DMCA etc.

The idea that a private TOS has the same level of public protection as copyright is downright childish.