Nagini, yes, this is at least a much better time to be playing than in Warlords, when the black pall of AoE from Graves and the Evul Undead covered everything. Blizzard dealt with that
pretty well in the adjustments they made at the start of Warlords, though I'm still a bit sad about Howl and Frog Kiss
et. al.
However, there are three current families of strategies that are pretty much pre-programmed. You learn the methods, and after that there is relatively little further thinking to do. In order of decreasing obnoxiousness:
1. Darkness. Keep Darkness up and use Nocturnal Strike as much as possible. This barely requires brain function.
2. Haunt and Stall. I'll quote Discodoggy's classic description from 2015:
The Haunt and stall strategy should be apparent. The plan is to start with the Ghastly Kid and use Haunt right away. If the pet that Haunt was cast on is slower than 305 speed, bring in the (Scalded) Basilisk (Hatchling) to use Crystal Prison, use Thrash once, then Feign Death to bring in the (Fiendish) Imp. The Imp can now force out the pet if necessary, which means that the enemy pet just took over 1000 damage, with the only damage given being whatever was done to the Ghastly Kid on the first round. By this time I’m usually so far ahead that finishing the battle is easy.
Nowadays we have the Wicked Soul and Teroclaw as superior alternatives, but the principle is the same. Haunt and Stall teams do have to consider how to manage cooldowns, maintain health, and keep the rotation going, so do require thought to run.
3. Clonedance. Set up a Cyclone, then swap and buff its hit chance and damage with Rain Dance or similar buff, and let the Cyclone do its thing. Clonedance can be countered hard by blocking or clearing the cyclone, or by swapping out the dancer, or by removing the cyclone, and there is considerable room to vary the team's composition by choice of a third pet. You can aim to make it all-out offensive with something like a Junglebeak or you can throw a stall or tanky pet in as your third with the idea of getting a second full round of Cyclone. And so I don't really think of Clonedance as objectionable in the same way as the others. Still, it is an out-of-the-box strategy, with a bit of DIY.
Bone Serpents are the biggest single-pet problem, because you can run multiples, and because they're too durable. Darkness birds can at least get taken down pretty quickly. Go out with 3BS and just press 1 2 1 3 1 each time, and you will beat most teams. That makes for a dull game, and as Rosqo points out, people never have to learn.