Emphasis on "old dinosaur"...? I don't think anyone's used a spinning disk seriously since Fizeau.
https://en.wikipedia.org/wiki/Fizeau_experiment Nipkow maybe, but not so much for sake of pulse generation.
Modern methods are fairly trivial, commercially speaking. Direct drive laser and AOM (acousto-optical modulator) are typical methods to get into the 100s of MHz, and GHz. Devices are readily available, including convenient modular formats like SFP.
But the quoted purpose is
LED drive, so lasers are N/A here.
As for the circuit, gate drivers should be fine. You might want individual resistors from each output, to avoid circulating currents between the two in case the propagation delays don't match. I wouldn't think a capacitor is needed, it should suffice to discharge the LED to 0V. Perhaps it's worth testing whether reverse bias clears charges from the junction faster -- in either case, the capacitor value can be adjusted freely, including infinity (replaced with short), so that still covers everything.
For still-faster drive, probably the next option is GaN transistors, but pretty soon at these time and impedance scales, you'll be hard limited by stray inductance on the PCB and into the LED chip itself (they're mostly/all wire bonded?). If you want to get
really fancy, you can consider peaking circuits (some series L and shunt C to somewhat compensate for the bondwire + pad + trace L), but that's going to get very tedious very quickly.
As for signal generation, buffering doesn't seem important, do the gate drivers have anomalously high input capacitance?
You say pulse is 10ns, but clock is labeled 100MHz which is a 5ns pulse.
How the oscillator works is important: if gated, you'll get a pulse train more or less as expected, give or take runt pulses at the start/end as the gate is asynchronous to the clock. If powered down, expect long startup time before the output is stable and consistent. It may be stuck high or low for some ~ms if it's a quartz type, and, I don't know about silicon oscillator types.
Using a latched clock gate, as might be constructed from some 74LVC or faster logic, would be fine, and then the oscillator can be free-running.
Doing both (a prompt start and accurate timing from an asynchronous trigger), is another matter. There are methods such as triggering an inaccurate but fast RC oscillator, then disciplining it (give or take acceptable phase/frequency error during the burst), or measuring the phase and disciplining to that fixed phase offset, etc. You might be better off buying test equipment if it needs to be this accurate.
Tim