Pat the chicken pieces dry. Mix the kosher salt, black pepper, and sweet paprika in a small bowl, then season the chicken evenly on all sides. Refrigerate uncovered for 1 hour to dry-brine.
Make the sauce while the chicken chills: In a saucepan, combine the ketchup, water, apple cider vinegar, brown sugar, Worcestershire sauce, yellow mustard, molasses, chili powder, garlic powder, onion powder, and cayenne pepper. Bring to a simmer over medium heat and cook, stirring occasionally, until glossy and slightly thickened, 10–15 minutes. Remove from heat, cool slightly, and divide into two bowls—one for basting and one clean bowl for serving.
Prepare a charcoal or gas grill for two-zone medium heat (about 375–400°F). Clean the grates, then oil them lightly with the vegetable oil.
Arrange the chicken on the cooler (indirect) side of the grill, skin side up. Cover and cook, turning once halfway, until the thickest pieces register about 150–155°F, 25–30 minutes.
Flip the chicken skin side down on the indirect side and continue cooking, covered, until the thighs/drumsticks are 170–175°F and the skin is rendered, 10–15 minutes more.
Brush the chicken with the basting sauce and move it to the hot (direct) side. Grill, turning and brushing a couple more times, until the glaze is set, lightly charred in spots, and sticky, 5–10 minutes. Avoid using the serving sauce on the grill.
Transfer the chicken to a platter and rest for 10 minutes so the juices settle.
Serve warm with the reserved clean bowl of sauce on the side.