Not quite, not all trainers have Pokemon that come out of Pokeballs.
In Ultra Sun/Moon when you speck/battle Hau at the ranch, his Pokemon is right next to him, so when he battles you, he does a hand gesture than his stage 1 starter Pokemon is in battle in front of him, the Pokemon also will do a battle gesture when the battle starts as well.
Pokemon breeders Pokemon all come out of friend balls, members of Team Rainbow Rocket have Master Balls, In the battle tree, some of the opponent's Pokemon Come out of great balls, and rarely ultra balls. But most people you face through the games will have Pokeballs, but it depends on where you are, and what kind of opponent you are faceing.
Source: Seen it happen with some opponent's Pokemon fighting Pokemon Breeders, Battling in the Battle Tree, And battling the heads of Evil teams In the Rainbow Rocket Episode.