I still have two Handyman push drills. Years ago I even modified a 1/4" hex drive extension to fit the Handyman chuck so I could use it with standard insert bits.We called those drills "punch drills", and used the heck out of them when I was a kid. Anyplace you would use a drill-driver now, we used those.
Ours were brass-plated steel, mostly, and had the storage for drills under the handle cap. The cap had a hole, and a ball with spring as a snap retainer. You pushed down the ball, and turned the hole to whatever size drill you wanted, and tipped up the drill to slide it out.
Drills had a notch on the shank, and had half of the end removed for 0.1" or so. The notch was a retainer notch, and the end took the drive. The nose of the drill had a sleeve that either pushed in, or pulled out, depending on maker, to let the drill slide in. When the nose sleeve was released, it moved a ball into the notch to hold the drill in place, if you had it rotated to the right position.
They worked pretty well, despite having drills that looked like a figure-8 in cross section, straight fluted, with a standard point.