I think a lot of the confusion in this discussion is coming from the specific example. The fine structure constant, an aspect ratio, or something like Reynold's

I think a lot of the confusion in this discussion is coming from the specific example. The fine structure constant, an aspect ratio, or something like Reynold's number for quantifying fluid flow turbulence -- these are all dimensionless numbers that scientists use, sure and there's probably nothing special about them. I think the confusion is in actually measuring these things. Scientists use dimensionless numbers all the time, but you have to directly measure quantities with units to indirectly get the dimensionless quantity.