We'd have every movement that could possibly benefit people infiltrated by the FBI. They would pose as Nazis during protests or try to instigate people to break the law in order to illegitmize the movement. Then we'd have a government controlled media only focusing what the provocateurs initiated.
Source? I know many cops and informers are actual nazis, and i've witnessed first-hand (in France) active collaboration between nazi groups and cops to repress demonstrations, but i'm not aware of cops pretending to be nazis to infiltrate social movements (where they would probably be kicked out for being nazis?!).
> try to instigate people to break the law in order to illegitmize the movement
Sure that happens, but that's a tiny phenomenon, except for anti-terrorist investigations where the so-called criminal activities are either legit protests (eg. pipeline sabotage) or plots outright invented by secret services (see also Haymarket affair, Eric MacDavid, COINTELPRO).
Many people are legitimately angry and want to burn institutions down. To be honest, i find that your line of thinking is more delegitimizing, because it suggests attacking the source of our problems and misery (eg. banks, homelessness) is immoral, and we should have hope in the institutions (who created this suffering in the first place and directly benefit from it) to solve the situation in a rational manner.
The law is a flexible thing based in a social context (and balance of powers), and if it was not for countless heroic acts of braving unjust laws, there would certainly have been no abolition of slavery, no anti-nazi resistance, no birth control rights... Not just in the USA, but pretty much in every country these progresses took countless acts of sabotage and resistance which were criminalized at the time.
They're discussing the infiltration of pro-Nazi demonstrations, which definitely did happen. COINTELPRO for example devoted 15% of it's resources to targeting White hate groups, leading to things like an FBI agent bombing a Baptist church and murdering a civil rights activist.
Good thing we don't have any of that though.