The Hammamatsu device has a 5 microsecond rise+fall time -> 200 kHz max freq. Thats probably on the high side of typical for a phototransistor. Also, it has zero sensitivity below 700 nm (where most visible LEDs have very little output). I need to do some spec searching for fast visible light phototransistors.
I tried replacing the LM741 op-amp with a much faster OP27. Then I tested it with that PT4415 based driver that I used on my desklamp retrofit (Nichia 219 Desk Lamp Retrofit) As I dimmed the driver, I could see a 112 kHz pwm rate.
Also, the phosphors used in white LEDs have some persistence. That could also limit the max PWM rate that one can see. I want to try driving a red LED (no phosphor) with a signal generator and see just how fast it can go.