Lather is not used to "treat" a strop. It is used to modify it's effectiveness. This seems to create confusion for beginner and intermediate practitioners alike. Lather is used to "treat" your strop the same way lather is used to "treat" your face before a shave. Then, allowing yourself to be corrected by someone telling you that a facial cream would be "better". You use the lather on your face and on the strop for the same reason, to improve your shaving experience. If you ask enough people eventually someone will point out that honing isn't really a good way to "treat" a razor.
Applying lather to a strop is designed to produce a greater stropping effect in less time. I would guess that about 60% of people that shave with a straight are using a poorly performing strop. A significant portion of those probably quit early in the learning curve because they are influenced to think that it is their technique that is the problem, not the equipment. This is true for shaving and honing, but much less so for stropping, which is a very simple act that relies almost completely on the surface action to produce the desired effect.