Remember that idioms do not have to express an idea or situation that is LITERALLY related to the words of the idiom itself.
Your first example, edited: 'The accident happened because the guy at the wheel drank too much.' (He was drunk.) In this sentence, you are talking about a car driver and a steering wheel. A better example would be a situation where no actual steering wheel was involved, e.g. : 'When we found serious errors in the data reported by the lab last year, we tried to determine who was at the wheel when those numbers were generated.'
Note that a very common form of this idiom is 'asleep at the wheel', meaning a failure to get something done, or to prevent something from happening, by not paying proper attention to the situation.
Your second example is almost ok, except for misuse of the word 'late' -- it is an adjective/adverb which must be paired with a noun: 'Due to my late arrival, I missed the boat for the position.'
Your last example does not quite work with 'the wheels fell off'. This idiom refers to a situation that was going ok for a period of time and then, unexpectedly, something went wrong, e.g.: 'Our team was in control for most of the game, but then the wheels fell off and we ended up losing 12 to 5.'
A good place to look up words and phrases used in current writings is --