That camera with 1/60 sync (and says ASA) must be at least 40 years old. :) More modern cameras commonly allow 1/200 second shutter with flash.

But this limitation (of not allowing flash with 1/1000 second shutter) is due to the type of shutter in the camera. It is not a property of the flash, electronic or bulb. It is a property of the camera shutter.

Focal plane shutters back in the 1960s were 1/60 second sync, but the focal plane shutters have gotten faster now, 1/200 second sync is common.

There is (or was) a FP sync flash bulb, called FP sync (for Focal Plane), which is a longer burning bulb, which allows any faster shutter speed. The issue is that when faster than sync speed, the focal plane shutter becomes a narrow slit moving across the film. At speeds less than sync speed, the shutter is fully open, which allows flash, even very fast flash. But at faster shutter speeds, it is just a narrow moving slit which is all that gets illuminated by the fast flash. Shutter speed is too fast to sync with flash.

The longer burning flashbulb allows any narrow slit to move, to occur at any point of the slit travel (the longer burning bulb will still be burning there). The faster shutter speed slit does of course seriously reduce the light, you have to open aperture to compensate for the lost exposure time.

In electronic flashes, there is a flash mode called High Speed Sync (HSS), which mimics the longer burning FP sync bulbs, to allow any fast shutter speed. The HSS flash "burns continuously" for the entire shutter slit travel time. It is NOT fast flash, the opposite, continuous like daylight, it cannot stop any motion. But it is High Speed Sync, meaning it does allow any fast shutter speed. Both the camera and the flash have to support HSS, and must be compatible with each other too.