Escape velocity (or even a velocity allowing a spacecraft to achieve low-earth orbit) is ideally achieved (as in, most efficiently) by lifting off exactly from the equator and rocketing straight eastward (zero azimuth). This takes full advantage of the approximately 1000 mph of initial “flinging force” supplied by the rotation of the earth around its North-South axis.
Polar orbit is relatively difficult to achieve since shortly after liftoff the vehicle needs to begin turning to the left toward a zero or near zero azimuth (or right toward 180), eventually spending/losing that initial flinging force. An large amount of extra fuel is therefore needed to be onboard at liftoff to continue propelling a (typically small, always unmanned) spacecraft (such as the Gravity Probe B) into a low-earth polar orbit.
The manned Apollo missions bound for the moon didn’t need to get into polar orbit per se, but the DID need to take somewhat of a hard left (azimuthal) turn after liftoff to get into (and supposedly temporarily DWELL IN) an elevated angle low-earth orbit, since it supposedly needed to staged in such a way to avoid going through the bulk of the Van Allan Belt.
Lighting up again to speed up to escape velocity, the vehicle needed to then simultaneously turn hard again, this time upward, from an orbital elevation of zero or thereabouts, initially probably to something like 50 or 60 degrees to avoid the Van Allen Belt, then quicky upwards of what, say 80 degrees elevation before straightening out toward its rendezvous with the moon.
So much extra fuel is needed for all of this maneuvering. How were multiple humans and all of their heavy spacecraft(s), science equipment, and extra fuel for the return trip, all lifted and carried all that way? Looking back now, it’s easy to see why LBJ and NASA might have started thinking about stealthily throwing in the towel and faking the whole thing.
An large amount of extra fuel is therefore needed to be onboard at liftoff to continue propelling a (typically small, always unmanned) spacecraft (such as the Gravity Probe B) into a low-earth polar orbit.
That's why the second stage is so much large than the first stage. Except that it's not.
You're really good at throwing jargon around, but you don't understand a syllable of it.
There are actual rocket scientists on this forum. I might be one of them (not saying).
You're out of your league.