You are right jumping off at LEO altitude would be silly. Which is why you would go to GEO then transfer to LEO. GEO is something like 35,000 km high. You burn

You are right jumping off at LEO altitude would be silly. Which is why you would go to GEO then transfer to LEO. GEO is something like 35,000 km high. You burn your rockets to fall towards low orbit, then pick up all the velocity you need falling towards LEO. Burn again once at perigee (or aerobrake) to steer your orbital path into circular rather than elliptical. A transfer from GEO to LEO is significantly cheaper than reaching LEO from the ground. Additionally transfer from GEO to anywhere else in the solar system is really cheap. It is hard to overstate just how much of an advantage a space elevator would be in reaching any location in the solar system including LEO. The only disadvange is travel time. Rockets to 300km get there quickly. Elevators to 35000km get there slowly.