Here are my crazy thoughts about your remarks, Jimv1 - beware spoilers!
Say, for example, your police department’s forensic expert is murdered and the evidence of remains of the biggest case ever are stolen, you’d think the police would check all available sources to see who was wandering about the building at the time - especially as even the cars have drones which can monitor a crime scene. Wouldn’t this cctv (enhance, enhance, left, 7, 8, 9) match the same person who is also allowed to later wander in and kill Lt. Joshi. Nastily.
That is oddly dropped, but maybe not. Joshi seems to be struggling under bureaucracy and almost doesn't seem as in control of what she is doing as you'd think. Remember the bit where she seems to have the ability to give K two days before she loses all control of what is going on? I almost felt like somebody was watching her as much as the police office was watching K's every professional move with all the mental monitoring, so even with being in the leadership she may not have had much power. The sway that the new replicant corporation wields may be hard to buck, like the rich family patriarch in Chinatown. And since the forensic guy seemed to be a replicant, I don't think anybody cared about him more or less (when he dies, his reaction seemed more robotic than human). They might have cared about Lt. Joshi, who was explicitly said to be human, but maybe the evil corporation was ahead enough and were about to get offworld in a way that was ahead of the investigation... going offworld might have been the plan to avoid the inevitable police investigation looking for Luv (or they might have offered up Luv as a rogue element if need be after getting Dekkard offworld).
Also, Luv was observing K and even rained explosive death on those scavengers. ‘Do your job. Find the child’.
K then goes straight to the Miracle Child, yet this lead isn’t followed up by those monitoring him. She’s still there at the end.
I think this is easy... they were looking for a male child since all the doctored paperwork led to the idea of the female child being dead, so I don't think they realized that the miracle was a she rather than a he. One thing that I keep lingering on - Joe "K" believed he was the child but then found out it was a she... but suppose that he was actually the boy child that they claimed died, but in reality he was the boy who had enough "replicant" to be thought to be a full replicant?
Perhaps the director was too busy thinking about how to squeeze another giant woman into the picture (Big Marilyn, we see you) ...or have I got this wrong?
The giantesses might have more point that critics have been giving them (other than being remarkable visuals). In the field of statues, which represented an older earth culture since it the remains of the old city before nuclear fallout, they were giant idealized beings who were unmoving images of unrealistic femininity. Contrast them to the world of 2049 where the uber-Joi Pink Giantess appears to Joe K in his time of uncertainty. She comes off first as an advertisement, but then she refers to our hero replicant as Joe. I think this might mean than Joi's personality transcended her programing and maybe is becoming just as aware as replicants are (though a pessimist might say that Joi's personality and love for Joe is all programming, but the Blade Runner films often explores the fine line between robotic intelligence and reality). Uber-Joi might be the dawning of a new awareness and individual lives for the holograms, and her liveliness and motion and confidence is far more real the frozen statues of Las Vegas even being an illusion.