Rounded terminals are older than type. Any letter painted with a brush or cut with a router ends in a curve, and signwriters have relied on that softness for as long as there have been shopfronts. In metal and phototype, though, rounding was a deliberate choice, and for most of the twentieth century it was a rare one.
The modern rounded sans is largely a product of the late 1970s. Helvetica Rounded appeared in 1978. A year later, Volkswagen commissioned a corporate face, and VAG Rounded — geometric bones, fully rounded ends — became one of the most recognisable brand typefaces of its era, later travelling well beyond the car company that named it. Arial Rounded followed the same recipe, and for two decades "rounded" meant exactly this: take a sans, round every terminal to the full stroke width, stop.
Screens changed the audience. Rounded sans faces read as friendly and approachable, which is why they became the default for consumer apps and children's products alike. Gotham Rounded, Nunito, Quicksand and Varela Round all arrived in the 2000s and 2010s, and the open-source ones spread everywhere Google Fonts did. Their shared weakness is the one VAG Rounded had: uniform rounding. Every corner is soft, so the I looks like a pill, the 1 loses its flag, and at text sizes the whole page goes slightly out of focus.
Vigil's answer is to ration the softness. Round where a stroke ends in open space and the eye reads it as a gesture. Round less where two strokes meet and the corner is doing structural work. Leave the letters that depend on a hard edge alone. The result reads as a clean sans at 16px and only becomes visibly rounded at display sizes — the reverse of what most rounded fonts do, and the point of the whole exercise.
The underlying skeleton is a geometric one, which is a category with its own hundred-year argument behind it — that history is here, and the practical guide to spotting and choosing one is here.