The girlfriend gave him very specific instructions about what to do if/when it happened, and reinforced them just before it happened with "Promise me that whatever happens, you will not let go".
I cannot imagine any scenario in which involving the authorities would have improved the situation and not violated her instructions, which he implicitly if not explicitly had agreed to follow. Most specifically "not let go", and getting her back to familiar surroundings, which per the account and others' accounts is important in stabilizing these events when they go that far.
The situation could have gone to hell, and one or both of them could have been injured or killed. Given that risk, his friend's "instructions" were plainly unworkable. If she had run in front of a car and been killed, for example, her estate might have come after him.
If she doesn't want the authorities to be involved, he would need at least a medical power of attorney. And even that would not not protect him from police responding to an apparent abduction. The only strategy to prevent that is to involve them from the start.
I cannot imagine any scenario in which involving the authorities would have improved the situation and not violated her instructions, which he implicitly if not explicitly had agreed to follow. Most specifically "not let go", and getting her back to familiar surroundings, which per the account and others' accounts is important in stabilizing these events when they go that far.