Cops will generally not detain someone in a case like this because fact of the matter is they don't have enough evidence to convict or hold a person. They didn't even have a crime. They had a missing person and a likely suspect not cooperating. And the window between the filed report and Brian's apparent last contact with his parents was only 3-4 days.
The problem with early detainment is it's easy to wiggle out of if you have good lawyer. They'd have him for 24 hours at most and he'd be out. And then the problem is you've flagged him as a priority suspect and drastically decreased his likelihood of cooperating and increased the probability of him fleeing.
When cops don't have enough to properly detain a person on bail, or convict, they're usually careful how they approach a situation. They want to give the suspect wiggle room that their behaviour and movements may lead to more concrete evidence. And they want to ensure when they do forcefully bring him in for questioning he is slow to seek legal advice.
The missing person report was filed, flagging the situation with the cops. Brian refused to cooperate. And only 3-4 days later he was "gone". That's a remarkably tiny window, before any body was found to further escalate the situation to what it is now.
Failure to appropriately monitor the suspect is another matter entirely.