Originally Posted by HTupolev

It makes sense as a very rough approximation across some common road cassette sizes.

200 pounds, plus say 20 pounds of bicycle, means that a ten pound difference is something like 230 vs 220. That's a 4.5% difference in gravitational resistance on the climb. In reality, the total resistance is going to change by a bit less than that because not all resistance while climbing is gravitational (i.e. there's still some stuff like aero drag).
Meanwhile the difference between a 24T cog and 25T cog is ~4.2%, or the difference between a 27T cog and 28T cog is 3.7%. So these 1-tooth gear ratio changes are in the same neighborhood as the difference in resistance caused by adding 10 pounds on a climb.

The approximation will break down with less traditional arrangements, like if the road bike in question is running a 1x setup with an 11-50 cassette. Then a 1-tooth difference is a 2% change to gear ratio, for instance.