The "cop-killer bullet" propaganda was started in the wake of Winchester's "Black Talon" development in the 90's, and "teflon-coated bullets" (KTW bullets) in the late 80's and early 90's.

First, the "Black Talon" nonsense. The Winchester "Black Talon" rounds were nothing more than a modern, reverse-tapered copper-jacketed hollow-point handgun round. You can buy their equivalent today, just under names that don't lend themselves to anti-gun propaganda. The Hornady Critical Duty rounds are very similar to the Black Talon bullet design. The black coating on the bullet was just a oxide-based coating (not Teflon) called "Luxalox," which is used to reduce barrel wear.

The "Black Talon" bullet wasn't designed to counter body armor. It was designed at the request of the FBI, who was still trying to come to grips with their pathetic performance in the 1986 Miami shootout, where eight FBI agents took on two criminals, and two agents were killed, and several more wounded. Here's the FBI recounting of this event:

https://www.fbi.gov/news/stories/fatal-firefight-in-miami

Out of this, the FBI blamed their weapons, which were .357 magnum revolvers[*]. The FBI sought several responses in their handgun purchasing to this event, and one of the responses was to have S&W create sidearms for the FBI chambered the new (then) 10mm Auto cartridge, which had been an idea brought forward by several private-sector gun people, including Jeff Cooper. Cooper wanted something that would bring most of the performance of the .41 Magnum revolver cartridge to a semi-auto, and the 10mm, as originally intended, did that.

But the FBI wanted more ballistic performance from their bullets too, and Winchester came up with the "Black Talon" line. Some other researchers, which included a police officer, had been working on a Teflon-coated bullet, which they branded as the "KTW" bullet. The KTW bullet was coated with Teflon for two reasons: a) it would help it penetrate car body panels, and b) it would help it penetrate the sloped angle of a windshield. The Teflon actually slows the bullet if it is fired against a Kevlar soft body armor vest.

So, the "cop-killer" bullets were research that was undertaken at the request or at the idea of law enforcement themselves, not the private sector.

What happened as a result?

The FBI found out that their affirmative action hiring practices put a whole lot of people in their ranks who can't handle a gun with some recoil, like the 10mm Auto. So they reduced the load in the 10mm Auto cartridge until their agents could handle it, and that was called the "FBI Load." S&W noticed that there was a lot of room left in the 10mm Auto case now, and if LEO's had trouble wrapping their hands around a double-stack auto, heck, why not just make a 10mm Auto "short" cartridge? S&W did this and it was known as the ".40 S&W," which many of us gun people called "40-short-and-wimpy" at the time.

The "Black Talon" ammo was pulled off the market, and uncoated bullets of a similar design were introduced in Winchester ammunition lines, and then several other bullet manufactures followed suite. Today, most self-defense ammo has bullets of very similar design to the Black Talon bullet, just without the Lubaloy coating which seems to cause people with journalism degrees to lose bladder control.

The KTW bullet was pulled from the market, for no good reason.

As for killing someone wearing a "bullet-proof" vest: most law enforcement personnel in the field are wearing NIJ Threat level II vests. A high-powered rifle with ball ammunition will defeat such a vest, easily. A broadhead from a compound bow will also defeat soft body armor quite easily. A modern crossbow will even penetrate IIIA level armor.

[*] In the hands of someone who knows what they're doing. the .357 magnum is still a superior handgun round, even after all the other rounds that have been developed in the last 90 years since the development of the .38/44 and resulting .357 Magnum, the .357 Magnum penetrates better than almost all other handgun rounds in common use. The FBI, being obviously ignorant of handgun ballistics, went on a wild goose chase at taxpayer expense, and lots of other law enforcement agencies followed them down the rabbit hole of trying to fix their marksmanship issues with magic cartridges and handguns, only to re-arrive where they claimed was "too weak" in 1986: A 9x19mm handgun, the Glock 17M. The FBI is still ignorant of handguns, and now they've just thrown in the towel and become most other law enforcement agencies. Which would I take? I'd take a S&W revolver in .357 over a Glock.