I do not understand the reluctance of crews to ask ATC assistance in such instances. "Excuse me old chap, but could we turn 90 left or right to lose a spot of height, don't you know? Perhaps even an orbit?" The neck hairs should be getting active in such a scenario. Consider the CALI crash. They were hot and high and should never have attempted what they did; a straight in NPA. That was a non-starter, so was this, so why get sucked up a cup-de-sac, only to go Oops later on. Pissing into the wind is something ill-advised. You should be able to realise it ain't going to work even before you start.