One of the things I was told as I was checking in at my residency program was that it is possible to be a "good resident" - manifested by being on time, having a good attitude, being an attentive mental sponge during cases, reading about your cases, working hard, volunteering/asking for bigger cases, honing your monkey skills, actively participating in journal clubs and PBLDs, all those things that are the hallmark of a good resident - and still fail the written exam.
About 20% of first time takers fail the ABA written exam. One in five takers are not *****s who flake on their basic case prep for three years.
It's probably enough to just learn anesthesia and be a good anesthesiologist. But a board failure is a life and income altering event. The written exam and ITE are nearly identical standardized exams, with all the quirks and minutia standardized exams have, and my personal opinion is that it's awfully bold (even arrogant) to blow off three free practice swings at the exam because you expect to be prepared after three years of being a good resident.
Train like you fight. Own the ITEs and the actual written exam will be a low stress formality.
Besides, however much it appears that your program doesn't care about your ITE score, people will notice and treat you differently if you score high. When you screw something up as a resident, it may be the difference between attendings thinking "oh he just had an off day" vs "lemme tell you what that bonehead did the other day" ...