The only application I've encountered where digital tools work better for me is having a DRO on a mill is extremely convenient.
Personally, i use microns instead of 0.001mm, too, when measuring that small. I forget the accuracy of my good calipers, but i could detect errors of around 2 microns if memory serves. It's been a long time since i cared about anything that accurate so i have two pairs of cheap plastic ones - scale and digital.
For 0.01mm there are only electronic or dial calipers.
Microns are the domain of grinding and lapping, you rarely ever need to go there with cutting.
I've already started calling the fab size of my processor as 80 angstroms (or whatever), so i probably back-propagated that.
Architectural rulers tend to divide inches into tenths for some reason. I have no idea why, because lumber comes in multiples of 12 (e.g. 8', 10', 12', 16') so if you design in multiples of 10 you're likely to waste a lot. If anyone knows I'd be curious to hear about it, mostly it makes drawing things to scale a pain in my ass..
If you're making a floor plan drawing at 1:100, a 240 inch wall becomes 2.4 inches on the drawing. The scales [of the drawings] and the scales [the tools] evolved together. (Similar to "why do computer people work in base 2 or base 16 so often?")
What "micron" are you referring to here? The "micron" I am familiar with is exactly that one (i.e. 1 micron = 1µm = 0.001mm).
To be clear, for a measurement where accuracy to less than 0.001" actually matters use a micrometer! Otherwise you're likely to screw up the part. But the advertised precision of 0.001" is totally repeatable within 0.0005".
EDIT: ah, I see your confusion. A "micrometer" is an aliased term. It means both "a millionth of a meter" and "a tool for measuring very precisely". I used the latter meaning above. Although in terms of order of magnitude precision they're identical--a micrometer accurate to 0.0001" is accurate to 2.54e-6m. It's possible to get within a handful of µm with decent calipers. Easier with a micrometer.