The push drills are often handier for overhead work, such as drapery hardware installation, but for Galoot style freehand drilling with small bits you can't beat an eggbeater drill. These drills are called double-pinion drills. An exception to this would be a drill like my Yankee 1446 which is a two speed and has two gear rings on the gear plate, an inner and outer ring. Instead of the idler gear, this drill has a small replaceable metal piece that is screwed to a post on the shaft of the drill just beyond the periphery of the gear wheel. The metal stop is just a millimeter or two higher than the upper surface of the rim of the gear wheel and is there to keep the gear plate from racking under stress.
Most decent eggbeaters are double-pinion drills, however.
There are typically two main types of chucks on eggbeaters. The predominate version is the one where you have 3 jaws that are conical sections. The taper on these match the inside of the chuck shell and they get squeezed together as you turn the chuck on the spindle. The important thing here is that they need springs to keep themselves evenly spaced around the interior of the chuck. If they are missing one or more springs, they may become slightly off center as you tighten the chuck and the bit may be misaligned as a result.