There's no need for the positions (or sizes) of the balls to be so accurate. The three-pins-on-three-pairs-of-balls arrangement is kinematic or critically constrained: there is exactly one position that will satisfy the system regardless of how accurately the contact components are positioned (as long as the errors aren't so egregious that they can't make contact at all). Reasonable errors in positioning the balls or pins relative to each other will cause an error in the position of the stylus relative to the probe body, but again: that is what the adjustment screws are for. The probe kinematics just need to put the stylus within a millimeter or so, and then you use the adjustment screws and a dial test indicator to adjust the runout (on a work probe) or perpendicularity (on a tool setter) of the stylus tip to as many decimal places as you want.
I probably could solder ball bearing balls made of other steel alloys, but those would be prone to rust and I do not want that on switch contacts. So, stainless steel seem to be the way to go.
Carbide is typical. Rigidity of the contact elements as well as the rest of the kinematic chain is more important to the performance of the probe than initial accuracy of the contact positions. Imagine jacking up a car: the wheel doesn't immediately leave the ground with the first crank, because the weight of the car has compressed the suspension and flattened the bottom of the tire against the road. So as you crank the jack up, you have to relieve that initial strain in the system -- the suspension has to fully expand, and the tire has to relax into its normal round form -- before it breaks contact. Switch between front and rear wheels, and it will probably take a different number of cranks, because the weight distribution and suspension characteristics (and maybe even the tires) are different, so there is more or less strain in the tire/suspension to relieve before the wheel leaves the ground. Whether the car is initially sitting on a flat surface matters a lot less (as long as it's not going to roll away on you...).
The same thing happens in a probe like this, although at much smaller scales. Under preload from the spring the mating surfaces of the pins and balls will conform to each other just like the tire conforms to the road, and the pins, probe stylus, body, etc all deflect a bit just like the suspension does. If there is more strain at one of the contact sets than the others, because of differences in spring pressure, or nonuniform rigidity in the system, this will result in different actuation travels in different directions. Ideally, the probe mechanism, not to mention the machine, would be perfectly rigid and any amount of deflection at the stylus tip would separate the probe contacts and be detected. That's not possible in the real world, but a hard material like carbide that resists deformation under point contact is preferable for the balls/pins because it means less overall strain, and less variation in strain with load/position changes. The rest of the system should be reasonably rigid as well, and most importantly it should be
uniformly rigid so that the amount of deflection required at the stylus tip to part the contacts is consistent in all directions.
It is possible to null out some residual errors via the chosen probing strategy or calibration method. During a measurement, a longer actuation travel in one direction makes the stylus appear smaller in that direction, so a strategy that compensates for runout compensates for that as well. A lot of that is baked into the standard probe/setter calibration routines from Renishaw et al, but getting the probe right to begin with so the error is consistent in all directions is of course better.
I can't help but think this physical electrical engineering problem is one that might be better fixed by using a non-contact detection method and some tweaking in software.
Physical contact is the simplest way to achieve these levels of precision and repeatability in tool and work probing. There are laser-based tool setters, but the optical and optoelectronic aspects required to get into the range of precision and repeatability that's required here are not trivial. They have advantages for very small tools (<1mm), but there's a good reason that 99% of tool and work probes use the basic mechanical contact mechanism the OP is trying to replicate. Work and CMM probes that can do surface profiling use strain gauges or capacitive sensors to provide an analog output, but they still use a stylus to physically contact the surface. Anything else is too vulnerable to confounding factors external to the probe. A work probe has to deal with all kinds of shapes and a wide range of materials -- steel, aluminum, plastic, composites, etc. A tool setter can count on dealing with things that are roughly radially symmetric, but still a range of materials -- tool steel, carbide, bonded abrasive, diamond, etc -- and it needs to accurately detect the very outermost edge of a thin, sharp cutting surface, often with complex facets. So all in all would be hard to come up something that would be simpler than mechanical contact unless you can constrain the material or shape quite a bit.