Stereotypes and prejudices persist because there a lot of people within any given group who conform to the stereotype. It's still prejudice though.
I'm ex police. I had rank so supervised a lot of officers. Young-in-service officers in frontline were actually among the most enlightened you'll find in society. Many of these male officers were doing school runs/child care/dentist visits etc far more than your average father. Partly because shifts enabled it but also because they wanted to - I know this because I was the person they had to ask if they wanted additional time off for a child-related reason (or any other reason for that matter). I didn't stand for misogyny on my time and very rarely saw it as a result.
I've seen some appalling misogyny in other areas of the force and the Crown Prosecution Service - all of which I've called out. Is the problem solved? No, of course not. There's still far too much. But it's got a lot better and its refusal to disappear completely sadly reflects the society from which police officers are drawn. You'll never get a truly equal police force until you have a society and government that is demonstrably pro women.
Some cops are good. Some cops are bad. The bad ones are always in the news/TV documentaries because that's what gets viewing figures up and suits the government narrative to keep wages low. No one ever reported on the number of times me and my team talked someone down off a bridge, took someone to safety, reassured victims and prosecuted abusers, comforted someone who'd just been told their loved one has died, administered first aid at a road traffic collision etc. All those qualities of patience and empathy are qualities I'd actually consider quite desirable in a date tbh.
I did not join the police to control people. Like thousands of other officers I joined to help people. I was a victim of domestic abuse in a previous relationship and it was life-changing. The police who came out to me were great and I decided I wanted to join to help others like me. What I and many others realised is that you can't reason with unreasonable people. You NEED to have the legal power to physically stop someone beating up their partner/child/random person on the street if they don't want to stop voluntarily. That's vastly different from joining up to deliberately lord it over others. I always used my legal powers very sparingly but absolutely had no hesitation when it was necessary (such as searching a domestic abuser's home for a hidden device on which he was tracking his victim).
All that said, I wouldn't advise anyone to date a police officer unless they're also one or in a similar high-stress role. My long, happy marriage would not have lasted had DH not also been in the job and understood why I regularly couldn't finish work on time for something important, why I often came home and couldn't really interact with anyone because of what I'd just had to deal with, why I'd lie awake at night worrying if impossible decisions I made were going to result in someone dying and me going to prison. That part of dating someone in the job is less attractive admittedly.
Recognition of red flags is important. Using someone's profession as a red flag is more akin to random gambling and not a smart strategy. The waitress test on your first date would be far more accurate.