The Pentium III does sound like a good chip choice for a retro cyberpunk story. Like they said in “The Matrix”, 1999 was the peak of human civilization.
(586 became Pentium, so 686 would be the Pentium Pro/II microarchitecture.)
The cartridge-style packaging of the Pentium II/III’s was also peak for the lineup.
My favorite PC I ever built was a dual-CPU Tyan motherboard that eventually held two screaming fast Coppermines. Needed a university copy of Windows 2000 to really make them sing—the Windows 95 series never supported SMP—and it was glorious.
I remember 1999/2000-ish I had both a Pentium III/Intel motherboard and Athlon PC. The Pentium III system was rock solid, and performed fantasic. Even the CPU and motherboard looked amazing.
The Athlon was solid but less reliable, various reboots and glitches. I kind have always had a preference for Intel since then.
A lot of that probably came down to the motherboard chipset. IIRC Intel made their own chipsets for the Pentium III and they were good and reliable. Athlons were coupled with chipsets from VIA and whatnot.
Some of those chipsets were fine and others were less reliable or compatible. The quality of the drivers for each chipset may also have mattered.
(586 became Pentium, so 686 would be the Pentium Pro/II microarchitecture.)