1. I would say that you deeply misunderstood the trolley problem if one person dead is the same as five, but more importantly,
2. I don't know if you've never been in a mental health crisis yourself, have never read any literature or listened to any mental health professionals on this matter, or simply missed that I also said "impulsive suicides", but access to firearms makes it so much more likely for somebody to impulsively kill themselves. I don't think this needs explaining per se, but you seem confused on this matter. Consider some of the most fundamental methods of intentionally killing oneself: high fall, hanging, stabbing, poisoning, and firearms. I emphasize when going over these that anything that introduces any sort of forethought/planning, latency, fear, or risk (even just a little) dramatically reduces the likelihood that someone will attempt suicide. I cannot express in words how much I discourage all of the following and want anyone to get help if they're struggling with this; this is just meant to be an objective analysis.

• A high fall requires you to find one first, and you need to think about one where you can guarantee you won't just break every bone in your body and be severely paralyzed for the rest of your life. It's also ingrained in the public consciousness at this point that you'll regret your decision the entire way down. It's also the case that someone will often sit around debating when they finally have everything they need to make the attempt, and for high falls, that can often mean being talked down and rescued if you're in an area with other people.
• Hanging often requires you to learn a specific skill in tying a noose (let alone for many people go out and buy some rope), and it means that you have to set up the rope, hanging location, and the object you'll stand on then kick away. Even then, there's a substantial fear that instead of snapping your neck, you'll hang there asphyxiating, fully conscious and waiting to die. There's also a fear that it won't work and you'll just fall or that – if you don't live alone – someone could intervene. After all, if someone finds a noose (let alone you tying one or setting it up), it's game over and you're going straight to involuntary hospitalization.
• Stabbing has easily accessible kitchen knives and tools, and you can guarantee death with reasonable certainty, so there's not much of a planning, latency, or risk aspect. However, you trade that for an enormous fear of the agonizing pain you'll be in until you pass out, taking this off the table for many people.
• Poisoning requires a ton of forethought and planning into how you'll go through with it, because even a cursory bit of research will show that just trying to OD on pills is one of the worst ways to successfully commit suicide (it's often a "cry for help" sort of attempt). You also have a fear of pain, and moreover, if you don't live alone, one that somebody will notice in time and get you to the hospital, where you could be left with permanent damage.
• Firearms (especially a shotgun) can fail but still have a very high success rate, and more importantly, it's completely immediate when done correctly. Fear is gone because death is immediate (even disregarding possible complications, that's what someone in this position would believe), and (assessed) risk is about as gone as it can be because there's minimal chance of intervention or failure. Let's now consider the other two: forethought/planning and latency. This method requires you to go through the complicated process of obtaining a firearm. If you've ever had a documented mental health crisis before, good firearms laws will restrict you from ever owning one or at minimum making it much more difficult to obtain. It also means you need to access the finances for a gun. And any good firearm regulation will introduce latency for exactly this reason: you need to wait a certain amount of time between purchase and obtainment exactly to deter impulsive suicides. So you need to find a place to purchase a firearm, have the funds, pick out a specific firearm from a selection of many, pass a background check, wait to receive it multiple weeks later during which at any point you can decide you don't want to go through with it and cancel (or someone else can see your mental health crisis deteriorating and intervene), pick it up, go home, and then still decide that you want to load it and pull the trigger on yourself. So as methods go, it's not ideal either. However, if the firearm is already in your home by the time this starts, then this calculus completely changes, and you suddenly have a method with as close to zero planning, latency, fear, and risk as possible, and that massively elevates the risk that someone will impulsively kill themselves.

Mental health intervention is important, but you can't intervene when someone is dead.