I think I sometimes hear sentences like "Much as I think ..." or "Soon as he ...". I'm not certain, but when the "as adj. as" form comes at the beginning of the sentence, people sometimes omit the first as in informal speech. Did I hear correctly?
There is a difference between the two usages you have identified:
Much as is a normal, correct, not informal equivalent of 'even though'. The speaker in the video says 'Much as I think he's a knob, I quite like working with Jeremy [Clarkson]'. There is no missing first 'as'.
It should not be, although it sometimes is, confused with the similar 'as much as something' used to compare two things or ideas and expressing them to be equivalent. For example I admire Michael for his good looks as much as for his integrity.
As soon as is an expression used to express that one thing follows another thing in time very soon, or instantly: As soon as the thief saw the police officer, he started to run. In casual or informal speech, the first 'as' is sometimes omitted.