Antennas and antenna theory generally all stems from the basic half-wave, center-fed dipole, probably the most basic of all antennas.
A quarter-wave vertical is only half of a dipole. It needs that missing quarter-wavelength to get back to its dipole roots. The radials take care of that. If the radials are a little longer than 1/4 wavelength, the ideal radiation pattern may be a little bit skewed, just as an off-center fed doublet's would be.
An end-fed 5/8-WL vertical, as Mr. Clean said, will have a considerably higher feedpoint impedance than a dipole or a 1/4WL vertical with radials. It will "work" only if this impedance is transformed to something close to the characteristic Z of the feedline, and that's what the matching system at the base does. The Ringo Ranger is a perfect example of that type of matching.
One thing a 5/8 WL vertical does NOT need is a ground plane/radials/counterpoise. Why? Because it's already over 1/2 WL. It's met the basic requirements. A half wave of input can be graphically overlaid on it, with a bit left over which increases the feedpoint Z. The matching system cancels that out.