In a large bowl, whisk together olive oil, cider vinegar, paprika, basil, celery salt, mustard powder, ground cinnamon, and red pepper flakes until well combined.
Season the chicken drumsticks with salt and pepper and add them to the marinade, tossing to coat thoroughly.
Cover and refrigerate for at least 2 hours, ideally overnight, to allow flavors to meld.
Preheat grill to medium heat (about 350°F / 175°C).
Remove drumsticks from the marinade and let excess drip off.
Grill the drumsticks for 35 to 40 minutes, turning every 7-8 minutes for even cooking and nice char marks.
Check that the internal temperature reaches 165°F (74°C) for safe consumption.
Remove from grill and let rest for 5 minutes before serving.
Serve hot with your favorite sides for a delicious barbecue meal.