Many popular first person games at the time were specifically tuned to extract maximum performance from Intel's pipelined floating point unit in drawing their 3D environments.
Since the K6-III inherits the same floating point unit as the K6-2 (low latency but not pipelined), unless the game was updated to use AMD's 3D-Now!
The base design was unchanged (the addition of SSE instructions was at that time of no performance significance) but Intel's new production process allowed clockspeed improvements, and it became difficult to determine which company's part was the faster.
Both firms were keen to establish a clear lead, and both experienced manufacturing problems with their higher-frequency parts.
For a time, the K6-III was a low priority part for AMD—something to be made only when all orders for high-priced Athlons and cheap-to-produce K6-2s had been filled—and it became difficult to obtain in significant quantities.
In consequence, AMD stopped making the K6-III in order to leave more room to manufacture Athlons (and K6-2s).
By the time the x86 CPU shortage was over, AMD had developed and released revised members of the K6 family.
These K6-2+ and K6-III+ variants were specifically designed as low-power mobile (laptop) CPUs, and significantly marked the transition of the K6 architecture (and foretold of AMD's future K7 project) to the new 180 nm production process.
Unfortunately, even with the 180 nm process shrink, the K6 architecture's short 6-stage pipeline while efficient, was difficult to scale with regards to clock speed.
As AMD's marketing resources at the time were focused on the launch of the upcoming Athlon K7 processor line, the 180 nm K6 series were relatively unknown outside of the industry.