The brief
A BMW with automatic push-button ignition came in that wouldn't crank, just a click or nothing when the button is pressed, the kind of thing that drives you up the wall when you're in a hurry. Push-button or not, the starter motor still does the work, and it wears out like anything else. The starter motor spins the engine over to start it, drawing a big current every time, and over the years the brushes wear, the bushes go, or the solenoid contacts burn. When it fails, you press the button and the engine doesn't turn, or only turns slowly. A worn starter doesn't recover, so it gets replaced.
The diagnosis
The check confirmed the battery and the charging were fine and traced the no-crank to the starter motor, worn out, which is why the engine wouldn't turn over on the button. The wiring and the ignition switch were fine. That's a starter motor replacement, with the battery confirmed good first, rather than guessing at a no-crank.
The work
The worn starter motor was removed and a new genuine BMW-spec starter fitted, the connections checked clean and tight, every fastener torqued to the manual figures. The battery and the charging were checked good, and the engine cranked and started cleanly. A test confirmed the engine cranking strongly and starting first time every time, no clicks, and no warning lights.
The outcome
The engine cranking strongly and starting first time every time, no clicks, no hesitation on the button, and no warning lights. The car went home starting properly again. A worn starter only gets less reliable until it leaves you stuck, so replacing it before that happens is the sensible move on a car with push-button start.