I too was in the Army (Missouri National Guard) and 300 yds was no easy task! Especially with an M-16 with open sights! Now, if I had a scope and I was comfortable with my rifle up to that yardage, I'd have no problem taking that shot. I can't say that I typically take that shot year in and year out,,, but I have before. I can also say I have missed from that distance as well.
I think it all comes down to what one person is comfortable with and to make sure that you are not just shooting to be shooting and hoping to kill the deer. KNOW you are going to make the shot before you shoot- then if you DO miss, then you know it was your breathing or flinching or something else....
...and on that day when my strength is failing, the end draws near and my times has come, still my soul will sing Your praise unending, ten thousand years and then forevermore!