Former military pilot here …
As Broomstick said, jet fuel is fancy kerosene. Taking a bath in the stuff, or more accurately, slogging around in a large mud puddle of kerosene-soaked dirt, would definitely trash a lot of skin for a few days.
And the fumes are kinda hard to breath. There’s a huge difference between you spilling a pint on the floor of your garage and an accident spilling a couple thousand gallons. The fumes from any petroleum product are going to be too dense to safely breath in that quantity. Fire Depts now wear oxygen tanks when beginning to deal with any large liquid spill for the same reason.
The military is also super-conservative about their own personnel safety outside of combat. They’d call in the guys in the moon suits when somebody spilled acid from an ordinary car battery in the auto-shop. This incident is probably mostly that attitude at work.
Hydrazine: The F-16 (which I used to fly), has a gizmo called the EPU or emergency power unit. It consists of a gas-pressurized tank with a gallon or so of hydrazine, a catalyst bed, and a turbine. The turbine is attached to a small electrical generator and a small hydraulic pump.
The F-16 only has one engine and hence does not have a lot of redundancy. The EPU was intended as an alternate source of electricity and hydraulic power to keep the airplane controllable if the engine quit, or if the main generator or hydraulic pump failed for any other reason.
Hydrazine was chosen (back in 1970) to power this thing because they needed something small and light that could be providing useful power within a second or so of getting the start signal.
On start, the tank valve opened, the gas pressure drove the hydrazine (a thick oily liquid), over the catalyst bed, wherein it reacted to form a high-temperature, high pressure gas, gaining volume and energy by a factor of a few thousand IIRC. That gas was directed to the turbine and then dumped overboard.
Dumping the gas overboard in flight wasn’t such a hazard, but if that thing fired off on the ground the exhaust plume was deadly out to a few feet, and noxious as heck for anybody within a couple hundred feet downwind.
A hydrazine spill was also dangerous, but nearly as much as the catalytic exahust, simply because the volume of vapor was so much less. Spilled on the ground, the stuff just sat there and evaporated like any other liquid. The fumes were still dangerous in concentration, but the area of concentration was much smaller.
AFAIK, the F-16 is the only aircraft of any kind in the US inventory that carries hydrazine.