Moshe, these planes are rebuilt by licensed Airframe and Engine Mechanics. If you are not licensed, you can't work on them. Okay, the engine caught fire. Mishaps occur in all kinds of aircraft, old and new. It's just a fact that can't be ignored.
When these planes are rebuilt to fly, they are torn down to the bare bones, and totally rebuilt. New wiring, hydraulics, rebuilt framework, and so on. So, technically, they are as safe as any other plane. But, things can go wrong, as in like any other aircraft, and they have to be certified flyable and inspected by the FAA in order to be airworthy.
Now, if it was just a "static display", which is a non-flyable aircraft, used only to exhibit, anyone can work on them. In fact, I am working on a fighter aircraft which will be a static display, and will never fly, but will (when finished) look like it can. It's like having a full sized model airplane. That's where we can use duct tape, and bailing wire.
Hubert