When you say "place the pieces on the grill", you seem to be missing the concept that a lot of kebabs aren't actually grilled on what we call a "grill" in America (and a lot of the world)... meaning a open surface like the one you posted an image of:
When I've seen "grills" they are simply open coal spaces with a rack for holding the skewers, so the meat is suspended directly over the coals and the skewers are required for holding them up. Here are some images of this sort of grill.
This one's a really old style:
The ancient Mycenaeans have a reputation as palace-builders and warriors, but they were also quite sophisticated cooks. More than 3,000 years ago, they used portable grill pits to make souvlaki and non-stick pans to make bread, new cooking experiments suggest. [...]
The Mycenaeans left behind amazing palaces and gold-littered tombs at sites like Pylos and Mycenae, but in these places, archaeologists also have found less glamorous artifacts, such as souvlaki trays and griddles made from gritty clays.
The souvlaki trays were rectangular ceramic pans that sat underneath skewers of meat. Scientists weren't sure whether these trays would have been placed directly over a fire, catching fat drippings from the meat, or if the pans would have held hot coals like a portable barbeque pit. The round griddles, meanwhile, had one smooth side and one side covered with tiny holes, and archaeologists have debated which side would have been facing up during cooking.
This one is mechanical and will actually turn the spits for you while it cooks:
So, as you can see, without the sticks, you'd be putting the meat directly in the coals, and no one wants that!