The trolley problem is wrong-headed to start with. Basically, it has two unlikeable options, and basically encourages psychopathic thinking.
I would disagree. It's a problem posed to challenge moral views.
Instead of asking "how can we save the five?" it wants to bully you into being a murderer through action or being a manslaughterer through inaction. And then, we're supposed to rationalize murder as being okay for the "greater good". But it's not okay. Nor is sacrificing a fat man (not even sure why that's morally different from pulling the lever, tbh).
Yes, it doesn't ask about how we can save the five. No, it doesn't want to bully you into becoming a murderer. It simply wants an answer. An answer that most people are reluctant to give and will try and sidestep around.
It wants to ask what it asks: Do you kill one to save five or do you not? That's it. At the fundamental level, that's all it wants to ask.
And a large majority of people don't seem comfortable confronting that question. So they like to ask things like this:
First of all, why doesn't the train have brakes? Or does it but they don't work?
Second, can you get off the train, and if so, maybe slow it down with your own death? Can you lay objects in front of the train? Can it be diverted to a third track? Can you warn someone that the trolley cannot stop? Are they really tied up (usually, they're just inattentive jerks and could be warned to move out of the way)? Can whoever tied them up be executed after rescuing them?
Nope, these questions are not addressed.
But these questions are the actual distractions to the problem. If you want, we can just be more explicit and construct a scenario where these things are impossible. We're behind a glass wall and can't interfere with the tracks or the trolley. The trolley is unmanned, and cannot be controlled remotely. Yes, 'tied up' actually means 'tied up'. The legal consequences are unknown to you and will be unknown to you until after the event is over.
But even in this scenario, I'm sure we can consider more things to do than simply pull the lever or not. However, the question remains; when all else is tried and all else fails, for one reason or another, the moment comes when we must make a decision. What do we choose then?
Now, shall we face the problem head on and decide, or shall we continue asking more of these questions in an attempt to dodge the real question that it's asking?