I have a mid-80s Karcher and a far older 2.2kVA generator that washed tractors for years. Admittedly the generator needed to be warmed up thoroughly and the governor working properly, and even then it bogged down on starting the washer, but it did get there. It's not about power, it's about the generator's ability to cope with the changing load without losing too much speed, before engine power has increased to compensate. Mechanical inertia (flywheel mass) and a fast- responding engine both help.
Our neighbour's generator had even less trouble doing it, despite a lower rated power - but then that was even older (i.e. more inertia), and was a diesel (where the governor is often better capable at this than a petrol governor).
There's also the tale of the Napier Deltic being used as a generator to drive AM radio transmitters. They replaced a Crossley engine that needed an enormous flywheel to maintain stable output, otherwise there was audible variation as it slowed down, whilst the Deltic (probably the fastest accelerating diesel ever) could simply ride a constant speed on its governor.