The phenomenon described in my post does work in both ways, different directions, amongst different groups, etc, yes.
However, what Hamas claims always spreads much, much faster than any other narrative. Social media is one aspect, but Western media also amplifies it with a caveat a lot of people don't seem to take into account, i.e., "Hamas run Gaza Health Ministry".
In the mix would be behaviours such as antisemitism and islamophobia.
A particular feedback loop is then created.
Media publications can also get things very wrong for various reasons which then worsens the problem. That's why Appeal To Authority is an important fallacy to keep in mind too.
Terrorist groups DO tend to be much, much faster at getting their narrative out than Sovereign Powers can. Governments are slower, especially as they have to investigate certain things. Whilst they're doing that, they're often tied up in Lawfare as NATO Strategic Communications Centre of Excellence made clear in their 2019 report which I've discussed previously in late 2023.
Hamas is what military analysts call a hybrid threat.
In the West, we should be more restrained and wait in order for the full extent of a situation to be ascertained. After all, we're not in immediate danger. But the psychology as mentioned above explains why many don't do this.
Hence, protests in major cities.