Dipole Antenna Current Distribution
It seems the confusion is due to the shield of the coax being "grounded".

Yes, this may be grounded at the transmitter end.  However, the antenna only "sees" the difference in potential between the two points it is fed at.  It doesn't matter to the antenna that you consider one of these to be "ground".

In the case of a true earth ground, it can matter to the radiation pattern when the antenna is less than a wavelength or above the earth.  This is why you often see a balun between a self-contained antenna like a dipole, and a non-symmetric transmission line like a coax cable.  This balun essentially becomes a single-ended to differential voltage converter at the RF frequency.

A quick hack that can still make a effective balun is to wrap the coax around something magnetic for a few turns.  This increase the impedance of the coax to common mode signals, while having no effect on differential mode signals.  At the transmitter, the shield is at a fixed voltage (ground) and the center conductor varies.  After the balun, both can vary up and down arbitrarily together, and only the difference between the two has low impedance and can carry any meaningful power at the RF frequency.