Push Pull setup for LED

Hmm. "real" electronic circuits do in fact have "inertia" caused by inductance and capacitance in the circuit, and apparently also the carriers in semiconductor junctions.

This is the "other" thing. High Speed LED and LASER drivers are "known technology", and instead of theorizing here amongst the amateurs, you should be looking up relevant technology that has already been developed. Probable search terms includ "LiFi" (like WiFi, but with light) abd "VLC" (Visible Light Communications), and LiDAR (like RADAR, but with light.)

The High-efficiency LED Driver for Visible Light Communication Applications

WikiPedia Article on Li-Fi

High-Speed LED Driver for ns-Pulse Switching of High-Current LEDs (paywalled)

Apparently the internal structure of the LEDs involved plays a significant role.

Most of the "High Speeds" being talked about are much higher than can be achieved with Arduino-class microcontrollers, like Perry's "80Gbps." Many of the relevant articles are in the range where you need very specialized equipment to measure the improvements of the fancy circuits involved.