Why does Bud White still investigate the Night Owl Murders even after he finds the perpetrators?

Your core assumption is wrong:

The perpetrators of the Night Owl murders in L.A. Confidential (1997) turned out to be three black people.

Nope. At the point—when Bud White is goes out to investigate the murders—the cops are explicitly being given details on suspects in the case and they are described as “Three Negro Juveniles” who were seen discharging shotguns in the air near Griffith Park and a make/model of the car they were driving (a maroon 1948-50 Mercury Coupe) is all they have to go on.

But the instructions of Captain Dudley Smith is pretty clear:

“48 two-man teams will shake three names apiece.”

Meaning the police have a vague lead to go on, and now they have a list of registrations/names compiled by the Los Angeles DMV and they are now all simply going out there to follow up on the leads.

That said after being given those directions, Ed Exley mumbles under his breath:

“Why not just put a bounty on them?”

Meaning, the instructions given seem to imply the “suspects” are actually the perpetrators of the murder and the cops should just go out and kill them without doing a real investigation into what happened.

But Ed Exley is motivated by justice and living up to the name of his father—who was a well respected member of the LAPD—and clearly is not out for blind vengeance. So the instructions given to him and other cops—which amounts to just “Shoot them first and let God sort it out.”—is not something he can easily accept or swallow.

And—as that briefing ends—everyone seems to go off in their own directions with some cops using the names from the DMV list while others follow their own instincts such as Bud White. We previously see him mentally connecting the dots between one of the victims of the Night Owl murders—Susan Lefferts—and a woman he saw a few nights before outside the liquor store sitting in the backseat of a car with two black eyes and—what appears to be—a broken nose:

“You got the wrong idea, mister. I’m fine.”

The conclusion is basically that while Bud White is portrayed as a “lunkhead brute” in the film who acts on raw instinct, he’s not necessarily completely brain dead idiot in his actions and behavior.

Bud has been shaken by the death of his former partner Dick Stensland and the death of a woman during the same crime who he spotted a few nights before—who has gone under some kind of plastic surgery to change her appearance enough to confuse her own mother—to motivate him to follow his own instinctual path in investigating the case.

His instincts coupled with Ed Exley’s desire to not swallow the “Three Negro Juveniles” as gospel is what propels the partners to finally work together in the end to get to the true—and extremely dark bottom of—who/what caused the Night Owl murders in the first place.