The surface plate. Modern precision all stems from the concept of a flat reference. However, the technique to make a surface as flat as possible (which is based on simple geometry) was first discussed in the early 19th c.
In terms of profound impact on modernity, the metal screw-cutting lathe.
I disagree. The modern (read:1800s) horizontal milling machine was really what enabled the industrial revolution. The kinds of parts you can make cheaply on a lathe is much less diverse than on a mill with proper tooling. Don't get me wrong, you need parts with circles but all the other shapes are what really enabled the industrial revolution. Mills and the complex shapes they can quickly create allowed us to cheaply crank out the tooling to make all sorts of things.
I'd argue firearms are the most important mechanical devices, as they and their production led directly to interchangeable parts, mass production, and precision tooling of all kinds. The modern world is literally impossible without firearms. And yes, you can make crude firearms yourself, but good ones require more skill and capability.
I think the idea of grinding three surfaces against each other to make them flat dates to the Sumerians and may have been independently discovered in pre-Columbian Mesoamerica.
In terms of profound impact on modernity, the metal screw-cutting lathe.