D&D 5e has intentionally been designed to limit the use of hordes of minions and servants and captive NPCs and such, to try to get back to the idea of a small party of adventurers out doing things themselves.
I recently concluded my last campaign of AD&D 1e. That party had gotten itself up to about 10-12th level, and was carting around a mass of 56 followers: friendshipped animals, pets, familiars, a djinn, and other minions, to supplement the 10 PCs -- it was quite a job tracking all that, and figuring out on any given task who would go do that specific thing, and what the rest of the entourage would do in the meantime. All of that was accumulated and facilitated by the AD&D 1e legitimate rules.
I started a new 5e campaign with the same players, and have had to be quite strict about the rules regarding animals and NPCs to avoid amassing such a menagerie. They keep trying, though. They're up to 6th level, and even with 5e and being strict, they've generally got horses for everyone (plus 2 to 4 spares), a mule and cart, a Found Steed buffalo for the Paladin, a befriended giant elk for the Druid, 4 war dogs, and a Familiar (usually an owl). A regular zoo, rolling down the roadways.
The skill of Animal Handling is not intended to be a magic portal to turn wild animals into servile companions; it's for dog trainers and cart drivers and horse riders to handle domesticated animals.
Speak With Animals is a magic spell that let you speak with animals. No more. In a situation like the last paragraph of your question, where you have a dog you've bee successfully using Animal Handling to train, and you cast Speak With Animals, I would let you choose whether to roll with Animal Handling or with Persuasion, whichever you choose, and give advantage to the roll due to the synergy of the two approaches. You would have a very good chance of getting this dog that you have a rapport with already, and now can communicate with clearly, to do exactly what you wanted. A wild giant poisonous snake, maybe not so much.
So yes, there are legitimate ways in 5e to get animal companions and servants and whatnot, but they are intentionally difficult to obtain and maintain, so as to preserve the feel and idea of Characters Doing Things by Their Own Abilities.