“Do Americans not own scales or something?”

For a good long while, no they didn’t. For a large fraction of American history a typical home kitchen had no bespoke measuring equipment at all; but tea cups, tea spoons and table spoons were typically available and made to pretty similar sizes, plus if you always used the same ones the proportions would be roughly the same, so meh.

A lot of traditional recipes were written this way, and it has remained so by tradition. A system of inexpensive, easy to manufacture measuring cups and spoons became standard equipment by the mid-20th century, and hasn’t changed to this day because it works just fine.

The US government is in the habit of publishing recipes with a deliberately low minimum equipment list. The United States Department of Agriculture for example conducts extensive testing on home canning recipes and methods, and deliberately writes their recipes to be used in poorly equipped kitchens, because the kind of folks who rely on putting up home grown vegetables for the winter don’t tend to spend a lot of money on Sharper Image kitchen gadgetry. Flipping through my copy of the Ball Complete Book of Home Preserving, I find about a third of recipes could be made using nothing but a mason jar or two as your only measuring tool, as most mason jars (excluding the deliberately decorative ones) have graduation marks in cups, ounces and milliliters molded into them.