How does a doctor's scale work?

The balance has a fulcrum very close to one end, so that the slide weight only needs to be a fraction of the items being weighed to be level. It is a matter of the moment of force being equal on each side of the fulcrum, so a tiny weight offsets the weight of a person.

The scale is simply a clever method of multiplying the length of that lever from Its fulcrum.

The fulcrum may be close to one end, but the person standing on the end moves their center of gravity further away. Could it be that the person straddles the fulcrum?

Could it be that the person straddles the fulcrum?

The fulcrum is up on the beam in front of your face. Your weight is pulling down on a hook almost right next to the fulcrum, exactly as you described with “the length of the end that the patient stands on would have to be tiny”.

But it gets more interesting. I can’t find a single picture that illustrates how the scale works. But I believe it’s like this:

A series of levers in the base translate your weight into a much smaller force. That force pulls down on a rod that goes up the column to another lever inside the horizontal case under the beam. That lever further translates the force, and it pulls down on the beam (with the hook right next to the fulcrum).

I get this from the pictures in this 17-page PDF:
Trouble shooting guide for Detecto Mechanical Physicians Balance Beam Scales

Thanks. At the very least, the scale is more complicated than just a single lever.

@LostInParadise “the scale is more complicated than just a single lever” not much, though.
It’s a lever with linkage on the short side to the platform you stand on. On the long side are the counterweights used to get the measurement. That’s it.