^It's like they say -- "The key is sincerity. If you can fake that, you've got it made." Unrealistic premises can be made to feel
convincing enough that the audience is willing to suspend disbelief. But that suspension of disbelief is not an obligation on the audience's part. It's something the storytellers have to earn. And if they fail to earn it, then viewers are entitled to complain.
In this case, the implausible premise of Castle helping out the police has been repeatedly justified over the show's history through the conceit of Castle's frienship with the mayor -- plus the fact that he's genuinely helpful to the police. So it's an implausibility that's been sold to us, that's sufficiently justified to earn our suspension of disbelief. They didn't just shove an unrealistic situation in our faces and demand that we accept it; they convinced
us to buy into it for the purposes of the story. But that was not the case here. They didn't justify the implausibility of a police officer ignoring the felonies being committed right in front of her. More likely, they didn't even realize that it was
a felony, so they didn't know there was anything to justify. Research failure is not the same thing as poetic license.