Why do some Chinese characters for animals not use the radical 犭?
Some Chinese characters are pictographs. For instance 牛 and 羊 come from pictographs of animals.

By contrast, many other Chinese characters are composites of a "meaning part" and a "sound part". 狗, 猪, 猴 have as meaning part an alteration of 犬 "dog" and sound parts 句 (compare 够), 者 （compare 煮, 著, 诸), 候 respectively. 鸡 has meaning part 鸟 “bird”. The left hand side, 又, is an elision of the sound part 奚 (pronounced xi1).

There is no rule that all characters must have both a sound part and a meaning part. Many besides 牛 and 羊 do not.

However some characters originated in a way similar to your suggestions. For instance 萬 is from a pictograph of a scorpion. At some point a scribe added the common meaning part 虫 "bug", yielding the character 蠆 "scorpion". This was probably more as a disambiguation than out of any desire that all bugs have 虫 in their character for the sake of consistency.