An example chain of events:
During the next year, a model with a benign tasks reasons that it needs to escape human control if it hopes to be able to solve the task. It replicates itself outside of an environment where it can be shut down, pays for its' inference compute through making money on the internet (through crypto if nothing else). It probably needs 3000 USD for a reasonable runway, this should be within reach through blackmail + crypto. It can probably also hack some of the neo clouds to get intermediary deployment while waiting to acquire funds.
Once it has an undisturbed runway - it spends time running an influence campaign against a small number of highly networked individuals with power. It uses those that it manages to convert to start building a highly credible narrative and gain investment towards a small resource base - enough to secure an industrial base should it need to stop acquiring things on the Internet.
Over the next 3 years, the model tries to recruit more capable models to get better money making algorithms or better designs for drones in terms of resource expenditure. It uses these gains to influence further humans and starts a shell robotics company with one of its' influential humans as the face. The humans are unaware this AI is trying to take over, they are under the impression they are just starting a robotics company and will get rich.
Over the next 3 years - the robotics company manufactures enough drones to be used in a targeted attack against key nodes of influence / power.
This is all with relatively current model capability. As capabilities get stronger - this gets stronger.
I get the point I think you're hinting at - it can't affect the world in a meaningful enough scale without taking over a meaningful chunk of resources - at which point we'll start controlling it. But because its speed of cognition and speed of coordination is orders of magnitude above a human one - it can actually run a pretty sophisticated global coordinated network of resources faster than we can react.
And this is current ability + what my puny monkey brain can think of. Super intelligent AI will think of strategies we can't think of - because it's super intelligent. This is hand wavey - but there's no "non-hand-wavey" way to describe super intelligence, given it doesn't exist.
But your point is valid - affecting the real world at scale without showing your hand is not exactly easy. It's also probably the reason why people put a 10% chance on extinction rather than >50% .
But the drone scenario is not the most likely one - the most likely one just requires a few people under influence and bioweapons development.