Prep the slow cooker. Lightly grease the insert or spray with nonstick spray.
Scatter the sliced onions on the bottom to create a flavorful bed.
Season the chicken. Pat the chicken dry and sprinkle with salt, pepper, smoked paprika, and onion powder. Place the chicken over the onions.
Mix the sauce. In a bowl, whisk together the root beer, barbecue sauce, garlic, and apple cider vinegar. Pour the mixture evenly over the chicken.
Cook low and slow. Cover and cook on Low for 4–6 hours or on High for 2.5–3.5 hours, until the chicken is very tender and easily shreds with two forks.
Shred the chicken. Transfer the chicken to a cutting board and shred with two forks.
Skim any excess fat from the sauce, then return the shredded chicken to the slow cooker and toss to coat.
Adjust the sauce. Taste and adjust seasoning with a pinch of salt or a splash more vinegar if you want extra tang. If the sauce seems thin, let it cook uncovered on High for 15–20 minutes to thicken.
Serve. Pile the saucy chicken onto toasted buns with coleslaw and pickles, or spoon it over rice, baked potatoes, or mashed potatoes.