Slaughter
Only: in all your appetite’s craving you may slaughter (animals) and may eat meat according to the blessing of the Eternal One your God that He has given you within all your gates; the tamei and the pure (alike) may eat it, as (of) the deer, so (of) the gazelle. (Devarim 12: 15)1
The argument for the “Oral Law” is that the Yisraelites would not know how to properly slaughter an animal if there was no “Oral Law” given to Moshe. We can see clearly in the very next verse that the only actual mitzvah related to proper slaughter is to spill the blood of the animal onto the ground like water.
Only: the blood you are not to eat, on the earth you are to pour it out, like water. (Devarim 12:16)1
We can also observe that we are forbidden to be cruel to animals. One such example we see in Shemot.
For six days, you are to serve, and are to make all your work, but the seventh day is Sabbath for the Eternal One your God: you are not to make any kind of work, (not) you, nor your son, nor your daughter, (not) your servant, nor your maid, nor your beast, nor your sojourner that is within your gates. (Shemot 20:9-10)1
We can also see from many other passages that animal cruelty is forbidden.* We can observe from these passages that it could be easily deduced that the quickest and least painful means of slaughtering animals is the proper way.
*Shemot 23:11; Vayikra 22:28; Devarim 22:4, 22:6-7, 22:10, 25:4
Eating Fat
The priest is to turn them into smoke upon the slaughter-site as food, a fire-offering of soothing savor – all the fat is for the Eternal One , a law for the ages, into your generations, throughout all your settlements: any fat, any blood, you are not to eat! (Vayikra 3:16-17)1
The argument for the “Oral Law” is that the Yisraelites would not know which fat is forbidden. However, we see from the preceding verses that the fat which is forbidden to us is already spelled out.
Then he is to bring-near from it his near-offering – a fire-offering for the Eternal One ; the fat that covers the innards and all the fat that is about the innards, the two kidneys and the fat that is on them, that is on the tendons, and the extension on the liver – along with the kidneys he is to remove it. (Vayikra 3:14-15)1
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1Everett Fox. The Five Books of Moses. New York: Schocken Books, 1997.