In my experience, Animal Handling is the best skill in the game and I will absolutely die on that hill. Like all skills it is very DM-dependent but across many DMs I have always seen this skill to be ridiculously overpowered.
The risk is generally low, you encounter an animal which was indifferent/hostile, and you attempt to improve its demeanor towards you. Risk is low because without a skillcheck it already doesn't like you, and if you fail, it still doesn't like you (similar to Persuasion). Reward is extremely high, because the upside is only limited by what the DM allows. I have seen this skill do anything from end/avoid combats to gain a powerful animal NPC companion usable in future battles.
Take for comparison a universally agreed-upon S-tier skill like Perception, which has also has low Risk but has lower Reward ceiling. With Perception or Investigation, the worst that may happen is you miss out on some treasure or secret door, or get ambushed by some enemy that surely you end up beating anyways. Potentially missing out on treasure is extremely impactful so I'd also rate these as S-tier, but even these are not as quite impactful as missing out on a powerful animal companion (even if only temporary)
