The efficiency of the brakes will come down drastically when the engine is off due to lack of vacuum and steering will become more hard too, but definitely you won't feel much at high speeds. The drive belt may be a cause for it as it drives the steering pump. This incident indicates the over speed of the car and the driver didn't have time to react, may be on a curve or due to some thing else. And most of the drivers will not admit their mistake and pass the blame to others or the car. Or in panic he might have locked the steering.
And definitely the minister and the staff deserve rewards for bravery and presence of mind, may it be for popularity also; few lives were saved
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Soon other ministers will learn swimming and train their staff as well (just joking)