The thing that separates a cheeseburger from a hamburger with a slice on top is the moment the cheese goes on. A slice of American cheese laid over a beef patty while it is still on the heat does not sit there: it slumps, runs into the seared crust, and seals to the surface so that meat and cheese stop being two layers and become one. The American slice is chosen for exactly this reason. Its formulation flows rather than splitting into oil and curd, so it coats the patty evenly and pools around the edges instead of sliding off in a sheet. That fusion is the whole sandwich. Everything else, the soft bun, the pickle, the squirt of yellow mustard and ketchup, is the cool, acidic frame around the one hot decision.
As a sandwich it works because the cheese does structural work as well as flavor work. A bare beef patty on a bun sheds juice into the bread until the bottom gives way; a lacquer of melted cheese on top acts as a partial seal, holding the patty together and slowing the bleed long enough to get the thing from the pass to the hand. The bun is deliberately soft and faintly sweet so it compresses to the patty rather than fighting it, and it is sized to the meat so the bread-to-beef ratio stays honest. The pickle and raw onion are not garnish: they are the sharp, crunchy counter that keeps a rich patty from reading as one heavy note. A flat-top cook drives a hard seared crust on the patty, and the cheese is timed to be fully molten at the second that crust is ready, which is the same patience problem a griddled grilled cheese poses, solved at higher heat and in less time.
The variations are a regional argument and each one deserves its own article rather than being crowded in here. The smashed style presses a loose ball onto the flat-top for maximum crust and melts the slice into a thin, crisp-edged patty. The Oklahoma onion build smashes shaved onion into that crust so it fries into the meat under the cheese. The Connecticut steamed version cooks patty and cheese together in a vapor box for something soft and loose rather than seared. The Juicy Lucy seals the cheese inside the patty so it arrives molten and dangerous instead of draped on top. The double, the bacon build, the green chile version, and the chain icons that standardized particular assemblies all keep the founding rule and change one thing, which is exactly the impulse that gave the cheeseburger its own name in the first place.