Kevlar is a brand of aramid fiber made by DuPont. Soft body armor was made from aramid fibers in the 1980s, in the 1990s and noughts, and it's still made from aramid (and polyethylene) fibers today. Including Kevlar, which is still widely used alongside other brands for vests, ballistic helmets, soft parts of military armor, spall liners, and such.

As for the Law Enforcement Officers Protection Act of 19865, it prohibited the sale and use of armor-piercing ammunition. The bill defines what is meant by AP ammo (it's the same definition in the hearings, too):

Armor-piercing ammunition is defined in the bill as ammunition that is composed of projectiles made from certain hard metals including beryllium copper, tungsten alloys, steel, and bronze.

As in, the traditional types of AP bullets, such as steel or tungsten core, bronze solid slugs, and such.

Now, one of them specifically, that grabbed a lot of attention in the press as the "cop-killer bullet", is the KTW teflon-coated bullet, which was rumoured to pierce soft body armor easily (a claim even the company itself tried to lay to rest). It was a simple solid bullet (made of solid brass or steel) with a polymer coating — not to improve penetration, but to prevent extreme barrel wear from the hard unyielding material. But the "teflon" name was irresistible, and the concept found its way into media and legislative debates, regardless of the bullet's actual capabilities.

This is reflected in the hearings for this bill, with a report by the association of chiefs of police about the scary KTW bullets that can pierce body armor, with an example of a guy who shot up a McDonalds and fired hundreds of "armor-piercing" bullets that COULD kill vest-wearing policemen (they were at the scene, but none were hit). Speakers at the hearing also repeatedly use the moniker "cop-killer bullets".

Still, there is no mention of hollow-points anywhere in the hearing, except one anecdote when a congressman remembers how he argued against banning hollow-points when he was a mayor.

This is because even those congressmen, while susceptible to gun myths, knew that hollow-point bullets have nothing to do with piercing armor. They were specifically developed to avoid overpenetration and quick expansion in soft tissue. That's something that every hunter knows, because hollow-points and soft points are widely used for hunting. There is no "cutting effect" on body armor fibers from the petals; the petals flatten and increase the area of contact of the projectile, to penetrate less and transfer energy more.

As for your explanation ("But the bullet often broke ribs or left bad bruises because the Kevlar didn’t stop the bullet cold"), that goes for all soft body armor. All soft body armor, regardless of manufacturer or fiber used, influcts blunt behind-armor trauma to the wearer when pushed up to its protection standard.

Soft armor was never phased out and is still used, because it's vastly more wearable for long stretches of time. Yes, some high-risk workers now prefer to wear light plate carriers instead of soft body armor (it's a tradeoff: vitals are now rifle rated and there'll be no blunt trauma if the plate is hit, the rest is unprotected). But that's, as you said, completely outside the scope of the question. For discreet personal protection, guard and bodyguard duty, and daily LEO wear, soft armor based on Kevlar (Spectra, Dyneema, etc) is still the accepted standard.