Exactly, or even without a history of mental illness they might ignore it if there's a reason to think the person might have intentionally cut ties (like, say, leaving home to follow the Grateful Dead, which might also imply addiction issues to LE since obviously there's a pretty big association between deadheads and various illicit substances). My cousin had not yet been diagnosed when he took off but we knew that he left his home willingly and had been in sporadic contact with friends and family before cutting us all off entirely, so no police agencies saw it as something they could help us with even though his last contact was telling us that government assassins were after him and he was going to disappear into the wilderness and live off the land (which would have been tantamount to suicide if he had followed through, since he had no wilderness skills and no gear and no money to buy gear with). But he wasn't directly threatening harm to anyone.
And the only PD that listened to us enough to tell us that was the city where my aunt lived (let's say Seattle; I'm going to change all the locations just to keep things a little anonymous) and my cousin had lived in before he left. His last communication for several years was from Denver. He called a friend to tell her that he was going into the mountains. Said friend lived in Chicago. So my aunt and the friend called Denver PD and explained the situation, and the response was, "Are you sure he was here? No? He's paranoid and obfuscating his whereabouts? What do you expect us to do?" Wouldn't even take a report because there was little reason to believe he was actually there (he was, we found out later). But they had no reason to believe anything untoward was actually happening in their jurisdiction so they couldn't take a report. That's something people don't understand--you don't always get to decide when/where a missing persons report is filed. You have to find a LEA that takes the case seriously enough and has jurisdiction to do something.
Sorry for ranting, but that black-and-white thinking bothers me too. I have a dear friend that I met during the whole thing with my cousin. She has a schizophrenic sister who refuses treatment and any help except the occasional cash gift. It's devastating to her when people judge her for "not helping" her sister--but there's literally nothing she can do. She can't force her sister to get treatment; there are no laws saying that mentally ill people are required to get help (aside from temporary commitments due to threat of harm or if they enter the criminal justice system and are ordered into treatment as a result of crimes they've committed; there are a few exceptional cases where people can be committed civilly but they're very rare and almost impossible), and it's not like you can just kidnap an adult and hold them against their will and force pills down their throats (that's a violation of a whole lot of laws itself). People who haven't lived it just don't understand, and are often very quick to judge unfairly.