The Ferengi were exploiting the entire population for personal gain by lying to them and manipulating their religion. Stopping them is the same as seeing somebody robbed on the street and stopping the robber.
The Prime Directive is a Non Interference policy that gets tapped for pre warp and post warp cultures alike.
It doesn't matter what the Ferengi were doing.
Janeway was not allowed to interfere in the ferengi's business just like she wasn't allowed to fracture those aliens religion by extracting their gods, or prove a hokum religion to be real which might prolong their dark age for centuries, because religion is a #### who hates science.
The Ferengi to her face argued the Prime Directive and Janeway was forced to submit to their arguments that she should follow her own laws, release the Ferengi and allow them to go back to manipulating the population down below.
Janeway admitted that the law was on their side.
Rather than saying "fuck the law" and tossing them in a cell, and going home, she felt obliged to beam them back planetside and then trick them into wanting to come with her because she thought that she was smarter than they were (Neelix in costume) and when that failed because she was not smarter than they were, she triggered that worlds apocalypse... I don't know if you've noticed but when ever we in the present on Earth reach some sort of deadline where the world is supposed to end, a few thousand idiots commit suicide and a handful of morons go on a killing spree to make the rapture easier for god, because gods job is hard enough as it is... One of the obvious ideas behind the Prime Directive is to avoid unforeseen consequences... What was more harmful to these people? Being turned into good Ferengi, or surviving the end of days?
As to your example.
1. the robber has mob connections and they kill your entire family to stop you testifying in court.
2. The robber is starving and his family starve to death waiting for "dad" to come back with a wallet.
3. The person being robbed is a billionaire and really doesn't care about the 200 bucks in his wallet.
4. You bean the robber in the nose, feel like a hero, but it turns out it wasn't a robbery at all... It's was a husband and wife roleplaying a sex fantasy.
5. You bean the robber in the nose, feel like a hero, but it turns out it wasn't a robbery at all... You had wandered onto a movie set and the producer sues you for 10 thousand dollars for interrupting filming and "damaging" an actor.
6. The robber kills you.
7. Trying to kill you, stopping you stop the robbery, the robbers bullets spray into the crowd and 4 children you do not know die.
8. You successfully stop the robbery, but you killed the robber in the process and you go to jail for 20 years.
9. You successfully stop the robbery, but you killed the robber in the process and you feel really bad about.
10. You successfully stop the robbery, but you killed the robber in the process and the robbers family sues you in civil court and you lose everything, and there's a lean on your wages for the rest of your life.
Janeway is not (only) a person, she is a representation of a foreign government. Everything she does to the Universe reflects on the Federation for the next ten thousand years. Your example is off slightly, since what happened each week on Star Trek was a Government (the federation) imposing, or trying not to impose it's values on other governments, which means for a real world argument comparative to the one you issued, you have to ask about how America is doing dip shit about China's occupation of Tibet... Or how China is doing dip shit about America's Occupation of Iraq... Other than harvesting their fertile oil fields of course.
If Governments disagree there is war.
The Prime Directive stops idiot Captains from starting wars with species they shouldn't too easily.
If Nagus Zek (while he was still in power) found out about what happened, he would have RAPED the Federation for such inviolate persecution of his people and the criminal activities of Janeway and the audacity of this Neelix person to think that he could impersonate the will of the Nagus.
Goods and services taxes would go up by 3 percent, and suddenly there's replicator rationing Federation wide.