I think there are a lot of standard (and wise) police procedures in place that are being questioned here. Police are run as a bureaucracy, just like hospitals, search and rescue organizations, the Red Cross, governments, etc, etc. They have to make sure they use their resources wisely, for the benefit of the whole community.
At the same time that San Jose police had this report of a woman in a rental car, in an emotional state who missed her flight, I'm sure they had dozens of other calls from other people who also desperately needed their help. Those other people's stories, of violence, drugs, crime, suicide, aren't featured on this forum, so they're treated as though they don't exist.
No organization can drop all their procedures and protocol because one particular case tugs at their heartstrings, no matter how desperate the family is to have them do it in their own case. Police can't say 'they seem like such a nice family and she must be a really special person, so we'll put her on the front burner and neglect all the less savoury, sad, desperate people who are our usual clients.'
In the real world, 'vulnerable' means children and other people who are living with a diagnosed vulnerablility, such as a senior with Alzheimers. Those are the people who get special police alerts.
Police can't just decide one person is vulnerable because her husband says so. In fact, I'm following a domestic violence trial on this forum of a husband who reported his wife to police as missing and suicidal, after she left him because he was abusive. It was all part of his campaign to discredit what she might be saying about him.
I'm sure if police could have located her, they'd have assessed her in person. But she didn't want to be located. Of course, they were quite aware that she may have been lying to them about being with other people, but if that's what she said, they have to take it at face value. Similarly, when the officer tried to elicit her location, she talked 'nonsensically', which, again, could have been a strategy to ensure he couldn't find her.
Would it have been appropriate for an ambulance to go trolling up and down every street in the neighbourhood, trying to find her? Then why do people believe police have infinite resources available to do just that?