> There's nothing unusual about private parties being able to assess penalties against other private parties
Absent a contract, there is, and even with a contract there are limits.
> Your bank does it all the time.
Within a contractual relation and governed by the contract, sure. But that's not what you are talking about.
> It does not make a mockery of the justice system.
Outside of the bounds of contract, it would, as it would amount to private parties making and adjudicating public law.
> If a failed frivolous lawsuit against Twitter automatically gave Twitter a claim on the plaintiff's assets, that would in fact not violate every tenet of the western justice system,
Yes, it would. Now, if merely a failed lawsuit did, it might not, as the failure itself is the conclusion of an adjudication, leaving no private determination to be made.
> It is a system that has obtained elsewhere and that people frequently advocate for.
No, its not, nor is it Twitter assessing a penalty. Loser pays is civil lawsuits is a thing, but it involves the court, not the offended party, assessing the penalty.
Loser pays for frivolous lawsuits only (but not all failed lawsuits) is also a thing, and is common in US jurisdictions, but requires a separate court determination that the claim was frivolous as well as the court assessing damages.
