This was not that.
This was a civil defamation case; the parents bought a case of actual material harm and harrassment of epic proportions before two seperate judges in two seperate states and both courts made the finding that Jones had indeed caused harm and harrassment .. and continued to do so over years.
If you mean higher bar for litigation, then maybe this lawsuit and its outcome shows that the bar isn't as high as you think when it comes to defamation?
To my understanding the case outcome is pretty much what I would expect, even considering the first amendment raising the bar. It's also interesting that there's been so many legal shenanigans in the case that it's hard to even keep track of them all.
That sealed the case outcome as, IIRC, at least one of the judges just ruled against them for not mounting any defence.
The first amendment does not protect you from the results of your speech, like someone deciding they don't like you because of what you said. That person is free to dislike you for what you said and the first amendment has nothing to do with it.
Similarly, if you say things that are untrue and cause damage to others, you may be held civilly liable for the damage if they sue you and convince a jury that you lied with knowledge and intent to lie. The first amendment has nothing to do with this.
This is the biggest difference - no one is claiming that all of the people who lost their loved ones in the 9/11 attacks were actually actors paid to pretend that they were grieving for their parents and children and friends. No one was encouraged to personally attack said victims and survivors to "expose their lies" because of 9/11 conspiracy theories.
Furthermore, defamation law works very differently for claims against public personalities ("Bush did 9/11!") compared to claims against private persons ("this random child shown crying in news reports after her classmates were supposedly killed is actually pretending!"). Also, vague accusations of orchestrating a criminal conspiracy / cover up are far harder to litigate than very clear claims of massive fraud. Finally, the Sandy Hook victims were generally able to show specific damages they suffered, attacks against them by people in their community, because of Jones' actions; Dick Cheney may have been more generally hated because of claims about 9/11 conspiracies, but was not directly harasses in the same way.
However, the first amendment is not absolute. Defamation is still a thing in the US. The first amendment creates a higher bar than many other countries (especially for public figures, but the victims in this case aren't public figures), but it is still possible.
If I were to bring a civil suit against you because the comment above offended my sensibilities, it would be quickly thrown out of court because it is your first amendment right to say anything you like, with certain exceptions that the government recognizes as limitations of this right.
Even though this is a civil matter, it is still a judgement on government law. This is not some contract dispute where the parties are simply seeking arbitration, with no government involvement except as a "service provider" for this arbitration.
He did take actions that, by civil law, created civil liabilities. He was sued over those liabilities. He failed to participate in the civil litigation process and lost badly as a result.
Civil and criminal law are not the same thing and your insistence otherwise doesn't change the reality.