Do you have a link to the clip? Was it in a humane dispatch scenario, presumably (i.e. not following up on a shot/wounded deer out in the countryside?).
First priority is human safety - that of you and anyone around you. Using an knife, potentially in hand to hoof (or antler!) combat with a distressed deer, is far less safe than using a firearm from a suitable distance.
Non-firearm methods of dispatch are discussed during HAD training, and a knife deemed an acceptable method for roadside dispatch in the appropriate circumstances - normally nothing to do with noise/disturbance but more relevant where there isn't a safe space in which to discharge a firearm (shotgun or rifle). An example could be a deer trapped in the front grill of a car that had subsequently been parked in a multistory car park (lots of concrete and other vehicles).
In this instance, the approved approach is to slice across the neck immediately below the skull, severing the main blood vessels - if you can do it by inserting then pulling forwards, great but it isn't the only way that works. The key is to do it like you mean it and not fanny around.
Hitting the atlas joint first time is not guaranteed even for the most experienced of us. It does not disrupt blood flow, only nervous tissue so you will still have a deer with a beating heart (the heart's pumping being intrinsic, not immediately linked to the nervous system).
Chest sticking - possibly, but that's a slower death and if you can see the chest to put a knife in, you can see enough to shoot it which puts you at less physical risk.
