Reaching orbital speed takes much more fuel than reaching orbital height
That’s not true. What do you reckon, that all the fuel is used for in the first second after ignition? Be it Saturn V or a Space Shuttle?
It’s used mostly for keeping the vehicle hanging in the air. It stays that way for quite a while. At least until the ascent is more or less vertical. That’s why the rockets take a more horizontal position as soon as possible, and then the remaining fuel is spent achieving orbital velocity,
The majority of the fuel is used merely for neutralizing the force of gravity.