I picked it up when i worked in the T.V. industry. I wasn't surrounded by the most inlightened people. but the phrase "screwing the pooch adds sort of a sense of humor to an otherwise very unhumorous mistake.
such as:
after blowing every single card in an uplink encoder
after dropping an Empeg
after waking up next to the waitress from the night before(ouch)
in short, nobody really screws pooches (well i guess there has to be someone somewhere, but i dont know them)
It's just a phrase i associate with botching things to an unfixable magnitude